In this research paper, I will derive the conditions for similarity between a triangle and its inscribed triangle. I will go through the inverse similarity conditions of an inscribed triangle for acute angled triangle and right angled triangle. I will use the basic laws of geometry to deduce the conclusion. Apart from mid-point triangle, we will analyze the conditions of similarity for other inscribed triangle.
When we select one point on each sides of triangle and join all those points, the triangle thus formed is called inscribed triangle. We know that mid-point triangle is similar to the triangle in which it is inscribed. Here, I will explain the existence of an inscribed triangle similar to a given triangle except mid-point triangle.
A. Inverse Similarity
Inverse similarity means the inscribed triangle is similar to the given triangle where corresponding vertices are not collinear.
On the basis of number of mid-point included by inscribed triangle, I will further go into detail.
As shown in figure 1, P and R are the mid-points of AB and AC respectively while Q is not mid-point of BC.
Condition when inscribed triangle includes two mid-points of a side of triangle
III. FOLLOW-UP RESEARCH
For obtuse angled triangle, is the necessary for the altitude of inscribed triangle to pass through the circumcenter of given center to be inversely similar to the given triangle?
Under what conditions of obtuse angled triangle, inscribed triangle will be inversely similar to the given triangle?
When the vertex of inscribed triangle includes the two mid points of sides of given triangle, then it is not possible for an inscribed triangle to be inversely similar to the given triangle without including the remaining mid points of a sides of given triangle. When the vertex of inscribed triangle includes only one midpoint of the sides of given triangle, then the inscribed triangle will be inversely similar to given triangle if and only if both inscribed triangle and given triangle are right angled triangle. For right angled triangle, inscribed triangle and given triangle are inversely similar if and only if the vertex of right angle of the inscribed triangle lies on the mid-point of the hypotenuse of given triangle. In case of acute angled triangle, inscribed triangle and given triangle are inversely similar if and only if one of the altitude of the inscribed triangle passes through the circumcenter of given triangle.
 He, G., 2015. San Jiao Xing De Liu Xin Ji Qi Ying Yong. Haerbin: Ha er bin gong. ye da xue chu ban she.
 Holshouser, A. and Reiter, H., 2014. Classifying Similar Triangles Inscribed In A Given Triangle. [online] Webpages.uncc.edu. Available at: [Accessed 14 June 2020]
 (not detailed). Retrieved from https://www.math.ust.hk/~mamu/courses/2023/W7.pd