Question

draw a triangle of the same area with 3 vertices on the circle​

Answer 1

Answer:

The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. The construction first establishes the circumcenter and then draws the circle. circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

Proof

The image below is the final drawing above with the red labels added.

Note: This proof is almost identical to the proof inConstructing the circumcenter of a triangle.

Argument Reason

1 JK is the perpendicular bisector of AB. By construction. For proof seeConstructing the perpendicular bisector of a line segment

2 Circles exist whose center lies on the line JK and of which AB is achord. (* see note below) The perpendicular bisector of a chordalways passes through the circle's center.

3 LM is the perpendicular bisector of BC. By construction. For proof seeConstructing the perpendicular bisector of a line segment

4 Circles exist whose center lies on the line LM and of which BC is a chord. (* see note below) The perpendicular bisector of a chordalways passes through the circle's center.

5 The point O is the circumcenter of the triangle ABC, the center of the only circle that passes through A,B,C. O is the only point that lies on both JK and LM, and so satisfies both 2 and 4 above.

5 The circle O is the circumcircle of the triangle ABC. The circle passes through all three vertices A, B, C

- Q.E.D

* Note

Depending where the center point lies on the bisector, there is an infinite number of circles that can satisfy this. Two of them are shown on the right. Steps 2 and 4 work together to reduce the possible number to just one