MathLabs

Problem 1

Let ABCABC be an acute triangle inscribed in a circle Γ\Gamma. Let A1A_1 be the orthogonal projection of AA onto BCBC, so that AA1AA_1 is an altitude. Let B1B_1 and C1C_1 be the orthogonal projections of A1A_1 onto ABAB and ACAC, respectively. Point PP is such that quadrilateral AB1PC1AB_1PC_1 is convex and has the same area as triangle ABCABC. Is it possible that PP lies strictly in the interior of circle Γ\Gamma? Justify your answer.
Step 5 of 5: The locus is the tangent line at D
Previous stepNext step
locus=tangent to Γ at D  ⟹  P∉int⁡(Γ)\text{locus}=\text{tangent to }\Gamma\text{ at }D\implies P\notin\operatorname{int}(\Gamma)
Detailed analysis

The line through DD perpendicular to the diameter ADAD is precisely the tangent line to Γ\Gamma at DD; every other point of this tangent line lies strictly outside Γ\Gamma. Hence the equal-area locus of PP found in Step 4 touches Γ\Gamma only at DD and otherwise lies outside it, so PP can never lie strictly inside Γ\Gamma. The answer is no.