in any triangle longer side corresponds to shorter median
Proof: Compute ma and mb using appollonius and equate.
2[ma^2 + a^2/4] = b^2 + c^2 __[1]
2[mb^2 + b^2/4] = a^2 + c^2 __[2]
ma^2 - mb^2 = (b^2 - a^2)/2 - (a^2 - b^2)/4 = 3/4[b^2 - a^2]
H.P.
Simon's factoring trick(complete the rectangle)
"Complete the Rectangle" , also called Simon’s Favorite Factoring Trick , is a clever algebraic method for factoring expressions of the form: x y + a x + b y + c xy + ax + by + c Or more commonly, you'll see it used in a simpler form: x y + a x + b y + d xy + ax + by + d But especially when we’re given something like: x y + a x + b y + a b xy + ax + by + ab It becomes very easy to factor. Let me walk you through it step-by-step. 💡 The Key Idea We add and subtract a constant to turn the expression into a perfect rectangle (a.k.a. a factorable quadratic or product of binomials). The “complete the rectangle” version of this trick usually works best on expressions like: x y + a x + b y + a b xy + ax + by + ab We treat it like this: x y + a x + b y + a b = ( x + b ) ( y + a ) xy + ax + by + ab = (x + b)(y + a) ✅ Step-by-Step Example Factor: x y + 3 x + 2 y + 6 xy + 3x + 2y + 6 Step 1: Rearrange the terms: Group like this: x y + 3 x + 2 y + 6 xy + 3x + ...
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