Online Geometry theorems, problems, solutions, and related topics.
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.
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.htmlPeter Tran
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 < DSimilarly AFLC can be shown to be collinear bisecting < A and < CSo the diagonals of ABCD AC and BD bisect the 4 angles and hence ABCD must be a tangential quadrilateral Sumith PeirisMoratuwaSri Lanka
To Sumith PeirisRefer to your solution1. 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 justifyPeter Tran
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,thenUKV and VGW are collinear but BUK and GWD need not necessarily be so. Hence my proof is fallacious Thanks PeterAntonio /Peter - any ideas as to how my proof could be corrected?