Prove that center of Wa , Wb , Wc are collinear. In tetrahedron ABCD, radius four circumcircles of four faces are equal. Let ABCD be an arbitrary quadrilateral. The bisectors of external angles A and C of the quadrilateral intersect at P ; the bisectors of external angles B and D intersect at Q.
We also consider a point R of intersection of the external bisectors of these angles. Prove that the points P , Q and R are collinear.
Prove that AC 2. AB BD. S is point that O is midpoint of HS. ABCD is a convex quadrilateral. We draw its diagonals to divide the quadrilateral to four triangles. P is the intersection of diagonals. Circles S1 and S2 intersect at points P and Q.
Furthermore, show that the points of concurrence are the same. Let CH be an altitude and CL be an interior angle bisector.
Show that the line joining the point of intersection of the tangents to the circle at the points C and D with the point of intersection of the lines AC and BD is perpendicular to the line AB. Triangle 4ABC is given. BHXC is a quadrilateral inscribed in a circle.
Let ABCD be a convex quadrilateral. MA ND Two circles intersect at two points, one of them X. The points A, B, C, D lie in this order on a circle o.
Given a triangle ABC. Let be given two parallel lines k and l, and a circle not intersecting k. Consider a variable point A on the line k. The two tangents from this point A to the circle intersect the line l at B and C. Let m be the line through the point A and the midpoint of the segment BC. Prove that all the lines m as A varies have a common point.
Let ABCD be a square. Given a triangle with the area S, and let a, b, c be the sidelengths of the triangle. Let H be the orthocenter of the acute triangle ABC. Determine the locus of the midpoint of the segment [P Q].
Show that XY is k to BC. MN BN 2 Let 4ABC be a traingle with sides a, b, c, and area K. Prove that the points A, B, K, L lie on one circle. Prove that the triangle is isosceles. PD 5 Prove that the line AP passes through the midpoint of the side CD. In the coordinate plane, any triangle congruent to triangle ABC has at least one lattice point in its interior or on its sides. Prove that triangle ABC is equilateral. Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
Let a triangle ABC. Let ABC be an equilateral triangle i. Sectors and Circles Problems. Problems, with detailed solutions, related to sectors and circle. A problem, with detailed solution, on a circle inscribed in one square and circumscribed to another, is presented.
A problem, with detailed solution, on a square inscribed in one circle and circumscribed to another is presented. Quadrilaterals Rectangle Problems. Rectangle problems on area, dimensions, perimeter and diagonal with detailed solutions.
Geometry Problems on Squares. Square problems on area, diagonal and perimeter, with detailed solutions.
Parallelogram Problems. Word problems related to parallelograms are presented along with detailed solutions. Trapezoid Problems. Trapezoid problems are presented along with detailed solutions. Solve a Trapezoid Given its Bases and Legs. Rhombus Problems. Rhombus definition and properties are presented along with problems with detailed solutions. Polygons Polygons Problems.
Problems related to regular polygons. Find the length of one side, the perimeter and area of a regular octagon given the distance between two opposite sides span. Problems related to parallel lines and alternate and corresponding angles. Compare Volumes of 3D shapes. A problem to compare the volumes of a cone, a cylinder and a hemisphere.
How to construct a frustum? If you cut off the top part of a cone with a plane perpendicular to the height of the cone, you obtain a conical frustum. How to construct a frustum given the radius of the base, the radius of the top and the height? Cone Problems. Problems related to the surface area and volume of a cone with detailed solutions are presented.
Pyramid Problems. Pyramid problems related to surface area and volume with detailed solutions. Geometry Tutorials Circles Parts of a Circle. Tangents to a Circle with Questions and Solutions. Inscribed and Central Angles in Circles. Definitions and theorems related to inscribed and central angles in circles are discussed using examples and problems.
Intersecting Chords Theorem Questions with Solutions. Intersecting Secant Theorem Questions with Solutions.
The properties of central and inscribed angles intercepting a common arc in a circle are explored using an interactive geometry applet. Adjacent angles: Two angles are called adjacent angles if they have a Common side and their interiors are disjoint. Linear Pair: Two angles are said to form a linear pair if they have common side and their other two sides are opposite rays.
Vertically Opposite angles: Two angles are called vertically opposite angles if their sides form two pairs of opposite rays. Vertically opposite angles are congruent. Alternate angles: In the above figure, 3 and 3, 2 and 8 are Alternative angles. When two lines are intersected by a transverse, they form two pairs of alternate angles. Interior angles: When two lines are intersected by a transverse, they form two pairs of interior angles. The pairs of interior angles thus formed are supplementary.
Is called a triangle. The three non-collinear points, are called the vertices of the triangle. Based on sides: Scalene triangle: A triangle in which none of the three sides is equal is called a scalene triangle. Isosceles triangle: A triangle in which at least two sides are equal is called an isosceles triangle.
Equilateral triangle: A triangle in which all the three sides are equal is called an equilateral triangle. Based on angles: Right triangle: If any of a triangle is a right angle i. Acute triangle: If all the three angles of a triangles are acute, i. Obtuse triangle: If any one angle of a triangle is obtuse, i. Altitude height of a triangle: The perpendicular drawn from the vertex of a triangle to the opposite side is called an altitude of the triangle.
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