Question

Geometry Question.

Answer 1

Side BC of triangle ABC is 4, side AB is 2*sqrt(19). Given that the center of the circle, passong though midpoints of sides of triangle, lies on the bisector of angle C.

Find AC

Answer 2

@ganeshie8

Answer 3

.

Answer 4

looks tricky...

Answer 5

Here is what I have done so far:
Let A_1, B_1, C_1 are the midpoints of BC, AC, AB. O is the center of circle. /_ACB=a. . As angle A_1C_1B_1 is congruent to angle ACB and is congruent to a, thus triangle A_1B_1C_1 is equal to triangle B_1A_1C, Consequently raduises of given circle and circle circumscribed around triangle A_1B_1C are equal.

Let line OC intersects the second circle at point M. MA_1=MB_1. OA_1=OB_1. If O and M are not same points, then OC _|_A_1B_1. CA_1=CB_1, AC=BC=4. in this case AC+BC=4+4=8 < 2*sqrt19=AB which is impossible. Thus M and O are same points.
Thus anle A_1OB_1 + angle A_1CB_1 = 180°. thus a=60. AC=x, by theorem of cosines x^2+16-4x=(2*sqrt19)^2 using this equation find that x=10, thus AC=10.

i'm i correct @ganeshie8 ?