The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + ve

Prove the following theorem: If the circumcenter of a triangle lies on a median of that triangle, the triangle is isosceles. Write the given and prove parts based on the diagram (see the attached file).

I need to see a construction and proof.
Let (triangle DEF) be equilateral triangle and Q is a point inside.
Prove that the sum of the three distances from Q to each side is equal to the altitude DD'.

1.Given: B is the midpoint of AC
BD is perpendicular to AC
Prove Triangle ADC is isosceles
(hint: first prove triangle CBD is congruent to triangle ABD)
The second part uses to same diagram
Given DB is perp to AC
AD is congruent to DC
m of angle C is 70 degrees
Find measure of ADB

1. Write a proof. AB*BE = CB*BD. Prove triangle ABC is congruent to triangle DBE.
2. A parent group wants to double the area of a playground. The measurements of the playground are width is 2W and the length is 2L. They ask you to comment. What would you say?
3. Find the length of the altitude drawn to the hypotenuse.

The included angle of the two sides of constant equal length s of an isosceles triangle is Z degrees.
(a) Show that the area of the triangle is given by A = 1/2s^2 sin Z
(b) If Z is increasing at the rate of 1/2 radian per minute, find the rate of change of the area when Z = pi/6 and Z = pi/3
(c) Explain why the rat

Please give a thorough explanation. Thanks.
1.Is it possible to have a triangle with sides measuring 10ft, 12 ft, and 23 ft.?
2.A mountain road is inclined 30 degrees with the horizontal, if a pick-up truck drives 2 mi. on this road , what change in altitude has been achieved?