Math Is Fun Forum

Find all values of p such that 2(x+4)(x-2p) has a minimum value of -18.
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Let \overline{PQ} be a diameter of a circle and T be a point on the circle besides P and Q. The tangent to the circle through point Q intersects line PT at R, and the tangent through T intersects \overline{QR} at M. Prove that M is the midpoint of \overline{QR}.
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In triangle ABC,

and

. Let

be the angle bisector of

.

(a) Prove that BC = BD = AD.

(b) Let x = BC and let y = CD. Using similar triangles ABC and BCD, write an equation relating x and y.

(c) Write the equation from Part b in terms of

and find r.

(d) Compute

and

using parts a-c. (Do not use a calculator!)

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Let A = (1,2), B = (0,1), and C = (5,0). There exists a point Q and a constant k such that for any point P, PA^2 + PB^2 + PC^2 = 3PQ^2 + k. Find the point Q and the constant k. What is the significance of point Q with respect to triangle ABC?

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Let A, B, and C be three points on the curve xy = 1 (which is a hyperbola). Prove that the orthocenter of triangle ABC also lies on the curve xy = 1.

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Let ABC be a triangle. We construct squares ABST and ACUV with centers O_1 and O_2, respectively, as shown. Let M be the midpoint of \overline{BC}.

unitsize(0.8 cm);pair A, B, C, M, S, T, U, V;pair[] O;A = (3,3);B = (0,0);C = (4,0);S = rotate(90,B)*(A);T = A + S - B;V = ro...

(a) Prove that \overline{BV} and \overline{CT} are equal in length and perpendicular.

(b) Prove that \overline{O_1 M} and \overline{O_2 M} are equal in length and perpendicular.

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Let P be a point inside square ABCD such that PA = 1, PB = 2, and PC = 3. Find \angle APB.

So how to do this one? I can't find any angles aaagggh

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In tetrahedron ABCD, \angle ADB = \angle ADC = \angle BDC = 90^\circ. Let a = AD, b = BD, and c = CD.

(a) Find the circumradius of tetrahedron ABCD in terms of a, b, and c. (The circumradius of a tetrahedron is the radius of the sphere that passes through all four vertices, and the circumcenter is the center of this sphere.)

(b) Let O be the circumcenter of tetrahedron ABCD. Prove that \overline{OD} passes through the centroid of triangle ABC.

Oh well. Thanks!