用微积分证明Euclidean Geometry定理
Theorems of Euclidean Geometry through Calculus
摘要：
我们重新推导泰勒斯，毕达哥拉斯，阿波罗尼乌斯，斯图尔特，赫伦，al Kashi，de Gua，Terquem，托勒密，Brahmagupta和欧拉定理以及内接角定理，正弦定律，外接圆半径，内半径和一些角平分线公式，通过假设存在一个未知的关系几何量之间的风险，观察如何关系的行为下，这些量的小偏差，然后自然地建立微分方程，我们就可以积分了。将通解应用于某些特定情形，给出了与期望定理相对应的特解。我们还建立了一个多项式方程和一组偏微分方程之间的等价关系。最后，我们评论了一个微分方程，它是在一个小尺度变换后产生的，应该涉及度量量之间的所有关系。
英文摘要：
We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.
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