Geometric Problem : Algebraic Proof - Relations Between Distances

Please see the attached file for the actual problem and a graphical illustration. Let ABC be a right triangle with sides a, b, and hypotenuse c. Let r be the radius of the inscribed circle and ra, rb, and rc be the radii of the escribed circles tangent to sides a, b, and c. What "RELATIONSHIPS" can you discover among the length ...continues

Regular Solids

For each of the 5 regular polyhedra, enumerate the number of vertices (v), edges (e), and faces (f), and then evaluate the quantity v - e + f. (One of the most interesting theorems relating to any convex polyhedron is that v - e + f = 2. -> This was given as part of the problem. I am not sure if it is of any value when solving t ...continues

Shortest segment

If P is any point on the parabola y = x^2 except for the origin, let Q be the point where the normal line intersects the parabola again. Find the shortest possible length of the line segment PQ.

Partial length of chord

I have a circle, diameter and radius unknown. It has a chord running through the center (point "O") and the end points are on the circle. Lets call this chord "SR" This chord SR bisects another chord "TU" and TU has midpoint "K". The only other information I have is that arc angle "TR" formed by point "T" from chord TU and point ...continues

Geometry : Probability that Three Points on a Circle will form a Right-Triangle

If n points are equally spaced on the circumference of a circle, what is the probability that three points chosen at random will form a right triangle? I know that for us to have a right triangle, the two points should form the diameter of the circle. What I have done is that I divided the problem into two sections. Section ...continues

Find the curve

Find the curve that passes through the points (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.

Line Intersections

Please see the attached file for the fully formatted problems. If P, Q are the midpoints of AB, CD respectively prove that i) AC + BD = 2PQ ii) AC+BC+AD+BD=4PQ ABCD is a quadrilateral. The midpoints of AB, BC, CD and DA are P,Q,R and S. Prove PQRS is a parallelogram. Find perpendicular distance from point... to line ...continues

A Combination of Algebra and Geometry : Grid Squares and Rectangles

Consider the following grid: 1) How many squares (of all sizes) are there in this grid? 2) How many rectangles (of all sizes) are there in this grid? 3) Can you generalize your results to an n by n grid? Can you generalize further? Please see attachment for grid.

Maltitudes, Circumcircles and Circumcenters

For any quadrilateral one can define the so-called maltitudes. A maltitude on a side of a quadrilateral is defined as the line through the midpoint of the side and perpendicular to the opposite side. Generally the four maltitudes of a quadrilateral are not concurrent, but if the quadrilateral is cyclic they are. Prove that i ...continues

Find the cross-sectional area and perimeter of a drain.

Water flows in a 400mm diameter drain to a depth of 150mm. Calculate the wetted perimeter of the drain and the cross sectional area of the water