all that it means by case 9iii is to use the picture as show

(all that it means by case 9(iii) is to use the picture as shown)

4. (See Figure 1.) Give the proof of Theorem 24 for Case (ii). Given: M and N are the midpoints of AB and AC, MX L AB, NX 1 AC, and X is on BC. To prove: X is on the perpendicular bisector of BC. Figure 1

Solution

According to the property of right angled traingle circumcenter (which is the intersection point of the perpendicular bisectors of a triangle) meets at the mid point of the hypotenuse

So,if we can prove X is the mid point of hypotensue BC then we can say POINT X IS ON PERPENDICULAR BISECTOR OF BC.

As point M,N are perpendicular bisectors of sides AB,AC IN TRIANGLE ABC ,So AM is equal to BM and AN is equal to CN ,So considering triangle NXC,MXB as NX is equal to AM which is equal to BM and MX is equal to AN which is equal to NC therefore according to pythogorus theorem CX is equal to BX ,Therefore X is the midpoint of BC which lies on perpendicular bisector of SIDE BC which is also the circumcenter of the triangle