Question

Draw a circle and a triangle with one vertex at the center of circle . draw another triangle of the same area with all three vertices on the circle ​

Answer 1

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


Regards,
Pratyush Purvarchan Upadhyay.

Answer 2

Given : a triangle inside circle with one vertex at center and two on circle

To find : Draw another triangle of Equal area with all vertex on circle

Solution:

Step 1 : Mark a point O

Step 2 : Take appropriate radius and open compass width and  draw  a circle taking center O

Step 3 : Take two points on circle A & B

Step 4 : Join OA , OB  & AB

Triangle AOB is constructed

to Draw another triangle of the same area with all three vertices on the circle

Step 5 : Draw a Diameter parallel to AB meeting  Circle at C & D

Now triangle ABC  & Δ ABD both will have same area as ΔOAB

as both triangle are on same base AB  & between two parallel lines

To get Δ ABC  join AC & BC

To get Δ ABD  join AD & BD

Area of Δ ABC = Area of Δ ABD = Area of Δ ABO

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