# Circle Tangency Criterion for Cyclic Quadrilateral
**Author:** Aviraj Das **Class:** 9th Grade student, West Bengal, India
## Abstract
We present a bidirectional characterization of cyclic quadrilaterals based on the tangency condition of two localized subcircles generated by diagonal intersections, orthocenter, and a circumcenter.
## 1. Introduction
The structural properties of cyclic quadrilaterals form a rich chapter in Euclidean geometry. This paper presents a necessary and sufficient condition for a convex quadrilateral to be cyclic: two constructed circles are tangent to each other at the diagonal intersection point.
## 2. Notations and Definitions
Let be a convex quadrilateral whose diagonals and intersect at .
- is the circumcenter of triangle . - is the midpoint of , and is the midpoint of . - is the circle through , with center . - is the orthocenter of triangle . - is the foot of the perpendicular from onto . - is the foot of the perpendicular from onto . - is the circle through , with center . - is the foot of the altitude from onto . - is the circumcircle of triangle .
## 3. Main Theorem
**Theorem.** The quadrilateral is cyclic if and only if the circles and are tangent to each other at .
In for