Problem 1

Let

ABC

be an acute triangle. Let

D

be a point on side

AB

and

E

be a point on side

AC

such that lines

BC

and

DE

are parallel. Let

X

be an interior point of quadrilateral

BCED

. Suppose rays

DX

and

EX

meet side

BC

at points

P

and

Q

, respectively, such that both

P

and

Q

lie between

B

and

C

. Suppose that the circumcircles of triangles

BQX

and

CPX

intersect at a point

Y\neq X

. Prove that points

A

X

, and

Y

are collinear.

Step 3 of 5: Power of A with respect to circle (DEX)

Previous step Next step

AD\cdot AS=AE\cdot AT

Detailed analysis

The circle through $D,E,X$ meets line $AB$ at $D,S$ and line $AC$ at $E,T$ . Computing the power of $A$ along each secant line gives $AD\cdot AS=AE\cdot AT$ .