perplexus.info :: Shapes : Vertices to incircle

Vertices to incircle (Posted on 2019-03-30)

Let A₁, A₂...A_n be the vertices of a regular n-sided polygon P. Let X be a random point on the incircle of P. Prove that

A₁X² + A₂X² + ... + A_nX² = nR²(1 + cos²(pi/n))

Here, R denotes the circumradius of P.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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The distance (AiX²) from point X to vertice Ai is igiven by the rule of cosine

AiX² = R² + r² - 2Rr cos(ai) where aiX is the angle XOAi, O is the centre of both circles, R is the circumradius and r is the inradius

A₁X² + A₂X² + ... + A_nX² = nR²+nr²- 2Rr [cos(a1)+cos(a2)+ ...+cos(an)]

The term with the sum of cosines is always 0 (with a square the cosines are opposed by pairs, with a triangle for a different reason). So:

A₁X² + A₂X² + ... + A_nX² = nR²+nr²

But r=Rcos(pi/n). So:

A₁X² + A₂X² + ... + A_nX² = nR²(1 + cos²(pi/n))

Edited on April 7, 2019, 4:59 pm

Posted by armando on 2019-04-07 16:57:57

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