Archimedes Laboratory Project™
Puzzles & Mental Activities that Enhance Critical and Creative Thinking Skills

Triangle ABC is isosceles (AB = AC) with an arbitrary point P on BC. Prove that the circumcircles of triangles ABP and APC are congruent.
Le triangle ABC est isocèle (AB = AC) ; un point P est placé au hasard sur BC. Montrer que les cercles circonscrits aux triangles ABP et APC sont congruents.
Il triangolo ABC è isoscele (AB = AC); su BC viene segnato, a caso, un punto P. Dimostrare che i cerchi circoscritti ai triangoli ABP e APC sono congruenti.

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Solution: Imagine ABC being any triangle with point P on AB. As O1 and O2 are the circumcenter of ACP and ABP respectively; it follows that triangles ABC and PO2O1 are similar. Here is the proof: AO2PO1 being a kite, then O1O2 must be perpendicular to AP. According to “circle theorem”: ∠AO2P = 2∠ABC = 2∠PO2O1, So ∠ABC = ∠PO2O1 By analogy, ∠O2O1P = ∠ACB This implies that: r1/AC = r2/AB Consequently, if AC = AB (isosceles triangle), then: r1 = r2 QED (see picture below) Please spread the word about the Sunday Puzzle!