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The Theorem of Apollonius (also known as Stewart's Theorem) states that in any triangle ABC as shown, m a 2 n b2 c CM 2 m n2 n m2 C b a I've never known this theorem, but it seems amazingly useful for calculating the length CM of a median of a triangle (if M is the midpoint of AB) based only on the lengths of the sides of the triangle. c A m n M B c=m+n Using the Theorem of Apollonius and AA' B' . We have the following derivation: e j 2 f a 2 A' B'r 2 e f 2 f e 2 2 d 2 d 2 d d d d j a d r2 2 2 2 4 2 4 2 d 2 d d ( j a 2 ) ( 2r 2 ) 2 2 2 2 d d2 j 2 a 2 2r 2 , but a2 2 2 2 2 so j 2r and j r 2 B' 2 j f a M r A s d B e k a Similarly, k 2 s and k s 2 2 2 A' In the figure above, A' B' d a 2 To finish the proof that point M lies on a lemniscate, notice that AA'BB' is an isosceles trapezoid (the proof is easy and left to the reader - look at congruent triangles within the figure and discover congruent angles that show j || k) Ptolemy's Theorem states that in a cyclic quadrilateral (such as AA'BB') the sum of the products of opposite sides equals the product of the diagonals. So we have: j k a a d (e f ) j k a2 d 2 r 2 s 2 a 2 2a 2 rs a2 2 Since r s is a constant, M must lie on a lemniscate. Q.E.D. I am grateful to Lockwood (1961) for hints to use Apollonius' and Ptolemy's Theorems for this proof. Lockwood, E.H. (1961). A Book of Curves. (p116). London: Cambridge University Press.