Two Triangles Centers At Play

Geometry Level 1

In triangle $ABC$ , $O$ is the circumcenter and $I$ is the incenter . $P$ is a point on the exterior angle bisector of $\Delta BAC$ such that $PI \perp OI$ . Extend $PI$ to intersect $BC$ at $Q$ .

Find $\dfrac{PI}{QI}$ .

Note : Figure not drawn to scale.

The answer is 2.

2 solutions

Bolin Chen
Mar 8, 2016

My solution is too much computation. I wanna find a more concise way.

Let $A(x_1,y_1)$ , $B(x_2,y_2)$ , $C(x_3,y_3)$ ,we can find the We can find the coordinates of O and I.(The expression is somewhat complex)

Then,find the linear equation of AP and PQ.And find the coordinates of O and I.And then calculate the length of PI and QI.

Could you find another way to solve it?:(

Your procedure is almost impossible to carry out.It is theoretically correct but unless we have numerical coordinates,it is not possible to practically solve using your method.

Manish Maharaj - 5 years, 3 months ago

So I am trying to find the better one= =...

Bolin Chen - 5 years, 3 months ago

A synthetic way to solve this is by butterfly theorem . I will let you consider this before I reveal the full solution.

Xuming Liang - 5 years, 3 months ago

Well solutions please ?

Arnav Das - 5 years, 2 months ago

What's the answer ?

Dristiron Saikia - 4 years, 8 months ago

draw a diagram to scale

Scott Rodham - 4 years, 7 months ago

Vishal Bansal
Nov 27, 2016

if you can imagine....it is an equilateral triangle

This is what I did, although I'd like to work it and get a more general answer

Andre Bourque - 2 years, 5 months ago

o,i coincides if you consider an equilateral triangle.

Balakrishna Padhy - 1 year, 1 month ago

Two Triangles Centers At Play

The answer is 2.

2 solutions

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