## Thursday, January 7, 2016

### Geometry Problem 1175: Six Tangential or Circumscribed Quadrilaterals

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1175. 1. This problem is almost identical to problem 883.
see link below for the solution.

http://gogeometry.blogspot.com/2013/06/problem-883-five-tangential-or.html

Peter Tran

2. BK must bisect angles < B and < K. KG similarly < K and < G and GD < G and D. Hence BKGD is collinear and bisects < B and < D

Similarly AFLC can be shown to be collinear bisecting < A and < C

So the diagonals of ABCD AC and BD bisect the 4 angles and hence ABCD must be a tangential quadrilateral

Sumith Peiris
Moratuwa
Sri Lanka

3. To Sumith Peiris

Refer to your solution
1. In general case diagonal BK of tangential quadrilateral BNKJ is not bisect angle B or angle D . Please justify for statement.
2. In my opinion B, K, G, D are not collinear as per your solution. Please justify

Peter Tran

4. I get your point Peter.

In a quadrilateral if the diagonals bisect the angles the quadrilateral is easily shown to be tangential with the point of intersection of the diagonals the centre of this in circle

The converse is not necessarily true

If for example U is the centre of circle within BNKJ, V of KLGF and W of GHDR,then

UKV and VGW are collinear but BUK and GWD need not necessarily be so.

Hence my proof is fallacious

Thanks Peter

Antonio /Peter - any ideas as to how my proof could be corrected?