perplexus.info :: Geometry : Take Two Incircles, Get Cyclic Quadrilateral?

Take Two Incircles, Get Cyclic Quadrilateral? (Posted on 2007-07-28)

Consider a triangle PQR whose incircle touches the sides QR, RP and PQ respectively at the points S, T and U.

J is a point which is located inside the triangle PQR such that the incircle of the triangle QJR touches QR, RJ and JQ respectively at the points S, K and L.

Determine whether or not the quadrilateral TULK is cyclic.

Submitted by K Sengupta

Rating: 3.0000 (1 votes)

Solution: (Hide)

TULK is a cyclic quadrilateral.

EXPLANATION:

We extend UT and QR to meet at H. In terms of the Menelaus' Theorem we then obtain:

(QH/HR)*(RT/TP)*(PU/UQ)= 1 .......(i)

Now, RT= RS; TP = PU and UQ = QS
Thus, (QH/HR)*(RS/QS)= 1, giving:

QH/ HR = QS/RS ..........(ii)

Again, JL = JK; QL = QS; RK = RS

From (ii), we now have:

(QH/HR)*(RK/KJ)*(JL/LQ)
= (QS/RS)*(RS/KJ)*(KJ/QS) = 1

Accordingly, in terms of Menelaus' theorem , we observe that H, K and L are collinear.

Now, HU* HT = HS^2, and HL* HK = HS^2, giving:
HU* HT = HL* HK

Consequently, TULK is a cyclic quadrilateral.

-------------------------------------------------------------

An alternate method has been posted by Bractals in this location.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Solution	Bractals	2007-07-28 22:09:13

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info