MathDB

1

Part of 2007 Postal Coaching

Problems(2)

sum |PA|^n + |PB|^n + |PC|^n + |PD|^n is constant wrt point P

Source: Indian Postal Coaching 2007 set 3 p1

5/25/2020
Let PP be a point on the circumcircle of a square ABCDABCD. Find all integers n>0n > 0 such that the sum Sn(P)=∣PA∣n+∣PB∣n+∣PC∣n+∣PD∣nS_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n is constant with respect to the point PP.
Sumgeometryconstantcircumcirclesquare
<APM + <BPC = 180^o , isosceles, < PAB =<PBC

Source: Indian Postal Coaching 2007 set 6 p1

5/25/2020
Let ABCABC be an isosceles triangle with AC=BCAC = BC, and let MM be the midpoint of ABAB. Let PP be a point inside the triangle such that ∠PAB=∠PBC\angle PAB =\angle PBC. Prove that ∠APM+∠BPC=180o\angle APM + \angle BPC = 180^o.
anglesgeometryisoscelesequal angles