Solve this problem Foundations of Geometry Plz Clear and det

Solve this problem (Foundations of Geometry)

Plz Clear and detail answer~^^

Given triangle Delta ABC, find a point X with AX^bar = BX^bar = CX^bar (so circle with center X and radius AX^bar goes through A, B, C) Does this construction work in neutral geometry?

Solution

Yes this construction work in neutral Geometry.

The above construction is valid only if ABC is a Acute Angled Triangle.

Here X is nothing but point of intersection of perpendicular bisectors of sides and is called circum center of triangle ABC

Explanation:

Given a circle passes through vertices of Triangle ABC ,it will be the circum circle off the triangle ABC.

Also if X is center of circum circle passing through ABC, X will be the circum center of triangle ABC.

Also given AX,BX,CX are radius of the circum circle .

Note:1) X lies inside triangle ABC if triangle ABC is Acute Angled Triangle.

2) X lies on midpoint of hypotenuse if triangle ABC is Right Angled Triangle.

3) X lies outside triangle ABC if triangle ABC is Obtuse Angled Triangle.