 GEOMETRY

From Wolfram MathWorld. An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon's sides. The center of the incircle is called the incenter, and the radius of the circle is called the inradius. While an incircle does not necessarily exist for arbitrary polygons, it exists and is moreover unique for triangles, regular polygons, and some other polygons including rhombi, bicentric polygons, and tangential quadrilaterals. Incircle and excircles of a triangle - Wikipedia. A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (JA,JB,JC), internal angle bisectors (red) and external angle bisectors (green). Circles - GMAT Math Study Guide. Definitions Properties of a Circle A circle is formed by an infinite number of points that are equidistant from a center. Brilliant Math & Science Wiki. Introduction How would you draw a circle inside a triangle, touching all three sides? It is actually not too complex. Simply bisect each of the angles of the triangle; the point where they meet is the centre of the circle! Then use a compass to draw the circle. But what else did you discover doing this? The three angle bisectors all meet at one point.This point is equidistant from all three sides. In order to prove these statements and to explore further, we establish some notation.

Let , and be the angle bisectors. Now we prove the statements discovered in the introduction. Brilliant Math & Science Wiki. A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. In this situation the circle is called an inscribed circle, and its center is called the inner center, or incenter. Imgur. Brilliant Math & Science Wiki. Brilliant Math & Science Wiki. In this section, we will try problems that can be greatly simplified by applying properties of the circumcenter, even though the problems themselves don't directly involve the circumcenter. Since the circumcenter is a rich structure that interrelates angles and lengths, using it correctly in a problem (e.g. International Mathematical Olympiad, or IMO) can be very powerful. For this reason, it is important to know how to spot circumcenters and the appropriate moment to use them. Brilliant Math & Science Wiki. If the altitudes of a triangle have lengths , then Triangle has area 15 and perimeter 20. Brilliant Math & Science Wiki. The area of a triangle, given the coordinates of its vertices, is equal to the absolute value of (The sign is positive if the points are given in clockwise order, and negative if they are in counterclockwise order.) Upon expansion, we get If the triangle is in three dimensions, then the area becomes Triangle exists in the 2D Cartesian plane and has two vertices at points and . Point exists on the parabola such that the -coordinate of point satisfies . Find the maximum possible area of triangle . We begin by plugging in our givens: Substituting and expanding, we get Using , the maximum occurs at . Area of Triangles - Heron's Formula Practice Problems Online. Brilliant Math & Science Wiki. Find the area of the triangle below. Imgur Since the three side lengths are all equal to 6, the semiperimeter is . Pythagorean Triples - Advanced. (You may like to read about Pythagoras' Theorem or an Introduction to Pythagorean Triples first) A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a2 + b2 = c2 Triangles And when we make a triangle with sides a, b and c it will be a right angled triangle (see Pythagoras' Theorem for more details): Brilliant Math & Science Wiki. Generating triples has always interested mathematicians and Euclid came up with a formula for generating Pythagorean triples. Primitive Pythagorean triples are Pythagorean triples and such that and are coprime. Note however that this formula generates all primitive triples but not all non-primitive triples. Euclid gives us the following formula: where and are any two positive integers that are usually called parameters. If we wanted to generate all the Pythagorean triples, we could then introduce a third parameter to our formula: Brilliant Math & Science Wiki. Cube. Cube From Latin: cubus - "cube, a die" Definition: A solid with six congruent square faces. A regular hexahedron. Try this Drag anywhere in the cube below to rotate it in any direction. Note how all the faces are squares and identical (congruent). A cube is a region of space formed by six identical square faces joined along their edges. From the figure, which one of the following could be the value of b ? : GMAT Problem Solving (PS)

Pythagorean Triples. A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a2 + b2 = c2 Example: The smallest Pythagorean Triple is 3, 4 and 5. Let's check it: Calculating this becomes: In the given figure, ABCD is a parallelogram and E, F, G and H are mid : GMAT Problem Solving (PS) Bunuel wrote: Attachment: SimilarTriangles2.jpg In the given figure, ABCD is a parallelogram and E, F, G and H are midpoints of its respective sides. How to Find the Area of Shaded Regions.