Problem 3

Let

ABC

be an equilateral triangle. Let

P

be a point on

AC

and

Q

a point on

AB

so that triangles

ABP

and

ACQ

are acute. Let

R

and

S

be the orthocentres of triangles

ABP

and

ACQ

, respectively. Let

T

be the common point of the segments

BP

and

CQ

. Find all possible values of

\angle CBP

and

\angle BCQ

such that triangle

TRS

is equilateral.

Step 3 of 8: Apply the symmetric lemma

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\alpha>\beta\implies RT>ST,\qquad\beta>\alpha\implies ST>RT.

Detailed analysis

Interchanging the roles of the two sides gives the second implication. If TRS is equilateral, RT=ST, so neither strict inequality is possible and alpha=beta.