MathDB

Equilateral triangle

Source: 2020 Taiwan TST

March 29, 2020

geometry proposedgeometry

Problem Statement

Let

O

be the center of the equilateral triangle

ABC

. Pick two points

P_1

and

P_2

other than

B

,

O

,

C

on the circle

\odot(BOC)

so that on this circle

B

,

P_1

,

P_2

,

O

,

C

are placed in this order. Extensions of

BP_1

and

CP_1

intersects respectively with side

CA

and

AB

at points

R

and

S

. Line

AP_1

and

RS

intersects at point

Q_1

. Analogously point

Q_2

is defined. Let

\odot(OP_1Q_1)

and

\odot(OP_2Q_2)

meet again at point

U

other than

O

.
Prove that

2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ

.
Remark.

\odot(XYZ)

denotes the circumcircle of triangle

XYZ

.

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