I am writing this article from the little experience & knowledge that I have gained from solving the RMO & INMO Geometry problems of our country. However, I have solved more than 7 problems in Geometry using this powerful method, which is completely based on similarity of triangles.
I am sorry that I could not post any topic due to my exams before.
A method related to similarity of triangles
We face several problems related to similarity of triangles. In some cases, similarity is used in proving theorems related to cyclic quadrilaterals. Now I come to my method.
Let ABCD be a quadrilateral whose AC & BD meet at O, such that ABO and BCO are similar. Then if we can write and in terms of where is the scaling factor of the two triangles.
The triangles are similar if:
1) It is a cyclic quadrilateral;
2) If its opposite sides are parallel , ie if it is a trapezoid;
In most problems, we are given that AC and BD are perpendicular, leading to the experessions of AB, CD, …..
Here I am representing a set of problems which I solved using the method:
1. (RMO-1992; Theorem 1 in my opinion ) Let ABCD be a cyclic quadrilateral such that ; AC meets BD at E. Prove that
where R is the radius of the circumscribing circle.
As is cyclic, (say).
Now, and .
Now from , Sine rule gives:
Comparing these yields the result.
2.Let AC and BD be chords of a circle with centre O such that the intersect at right angles inside the circle at point M. Suppose respective midpoints of respectively. Prove that is a parallelogram.
(Sorry for my delay in updating)
Now as ABCD is a cyclic quadrilateral.
AS ABM is a right triangle,
AS CDM is a right triangle,
Hence in and ;
Now; triangle AMK gives:
From the problem before, we know that
So we have;