# dialectics

title: Example Note
created_at: 2026-06-16 11:35:33
content_json:
  [{"id": "block_zcch6vd", "side": "left", "html": "<p>We are given a right-angled triangle as an independent object.</p><img src=\"data:image/png;base64,[encoded data, 24176 characters]\" alt=\"Canvas Drawing\" width=\"400px\" style=\"width: 400px; height: auto; cursor: pointer;\"><p></p>"}, {"id": "block_7cw9fbh", "side": "right", "html": "<p>Squares can be built on the sides of this triangle.</p><img src=\"data:image/png;base64,[encoded data, 23892 characters]\" alt=\"Drawing\" title=\"Canvas Drawing\" width=\"150px\" style=\"width: 150px; height: auto; cursor: pointer;\" data-fabric=\"[encoded data, 19612 characters]\"><p></p>"}, {"id": "block_1zfm7w0", "side": "left", "html": "<p>Area difference of squares.</p>"}, {"id": "block_p46ya0e", "side": "right", "html": "<p>In this construction, the triangle as an independent object disappears and becomes only the difference in the areas of the two squares. This creates a contradiction between the triangle as an independent object and the difference in the areas of the squares.</p><img src=\"data:image/png;base64,[encoded data, 22160 characters]\" alt=\"Drawing\" title=\"Canvas Drawing\" width=\"150px\" style=\"width: 150px; height: auto; cursor: pointer;\" data-fabric=\"[encoded data, 5284 characters]\"><p></p>"}, {"id": "block_qvc8djg", "side": "center", "html": "<p>This contradiction is overcome in the form of the Pythagorean theorem, where the length of the hypotenuse is converted into an area (square) and then equated to the sum of the legs, which are also converted into areas.</p><p></p><p>This can be written as follows. There are two ways to find the area of a large square:</p><p></p><ol><li><p>1.<span latex=\"S_{\\text{first method}} = (a+b)^2 = a^2 + 2ab + b^2\" data-type=\"mathNode\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>S</mi><mtext>first&nbsp;method</mtext></msub><mo>=</mo><mo stretchy=\"false\">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">S_{\\text{first method}} = (a+b)^2 = a^2 + 2ab + b^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em; vertical-align: -0.15em;\"></span><span class=\"mord\"><span style=\"margin-right: 0.0576em;\" class=\"mord mathnormal\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.0576em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">first&nbsp;method</span></span></span></span></span></span><span class=\"vlist-s\">\u200b</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">a</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em; vertical-align: -0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em; vertical-align: -0.0833em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">a</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em; vertical-align: -0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\">ab</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">b</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>2. <span latex=\"S_{\\text{second method}} = 4\\left(\\frac{1}{2} \\cdot ab\\right) + c^2 = 2ab + c^2\" data-type=\"mathNode\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>S</mi><mtext>second&nbsp;method</mtext></msub><mo>=</mo><mn>4</mn><mrow><mo fence=\"true\">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>\u22c5</mo><mi>a</mi><mi>b</mi><mo fence=\"true\">)</mo></mrow><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">S_{\\text{second method}} = 4\\left(\\frac{1}{2} \\cdot ab\\right) + c^2 = 2ab + c^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em; vertical-align: -0.15em;\"></span><span class=\"mord\"><span style=\"margin-right: 0.0576em;\" class=\"mord mathnormal\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.0576em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">second&nbsp;method</span></span></span></span></span></span><span class=\"vlist-s\">\u200b</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2em; vertical-align: -0.35em;\"></span><span class=\"mord\">4</span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(</span></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"></span></span><span class=\"\" style=\"top: -3.394em;\"><span class=\"pstrut\" style=\"height: 3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">\u200b</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em;\"><span class=\"\"></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">\u22c5</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mord mathnormal\">ab</span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)</span></span></span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">c</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em; vertical-align: -0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\">ab</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">c</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><p></p><p>This gives:</p><p></p><p><span latex=\"a^2 + 2ab + b^2 = 2ab + c^2\" data-type=\"mathNode\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">a^2 + 2ab + b^2 = 2ab + c^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em; vertical-align: -0.0833em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">a</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em; vertical-align: -0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\">ab</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">b</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em; vertical-align: -0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\">ab</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">c</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p></p><p>Shorten it:</p><p></p><p><span latex=\"c^2 = a^2 + b^2\" data-type=\"mathNode\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">c^2 = a^2 + b^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">c</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em; vertical-align: -0.0833em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">a</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">b</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span></p>"}, {"id": "block_34h7kn8", "side": "left", "html": "<p>The Pythagorean theorem resolves the contradiction between a right triangle as an independent object and other geometric objects.</p>"}]


