Problem 4

In triangle

ABC

the bisector of angle

BCA

intersects the circumcircle again at

R

, the perpendicular bisector of

BC

P

, and the perpendicular bisector of

AC

Q

. The midpoint of

BC

K

and the midpoint of

AC

L

. Prove that the triangles

RPK

and

RQL

have the same area.

Step 3 of 4: Deduce P and Q are symmetric on chord CR

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In plain words

The perpendicular from the circumcenter $O$ to chord $CR$ bisects both the chord $CR$ and the base $PQ$ of the isosceles triangle $OPQ$ .

OE \perp CR \implies CE = ER \text{ and } PE = EQ \implies RP = CQ,\ RQ = CP

Detailed analysis

Drop the perpendicular $OE \perp CR$ from the circumcenter $O$ . Because $CR$ is a chord of the circumcircle, $E$ is the midpoint of $CR$ ( $CE = ER$ ); because $OP = OQ$ , $E$ is also the midpoint of $PQ$ ( $PE = EQ$ ). Subtracting and adding along line $CR$ gives $RP = CQ$ and $RQ = CP$ .