Geometry for Students and Parents. Sergey D. Skudaev

BE/2. See figure 14.

      Figure 14. Area of a triangle.

      Bisectors

      A line drawn from a vertex of a triangle that divides the angle of origin into two equal angles is called the angle bisector. See figure 6. Each point on an angle bisector is equidistant from the sides of the angle.

      The bisectors of a triangle are concurrent. A point of their intercept is called the incenter. The incenter is equidistant from all sides of the triangle and is the center of the inscribed circle. See figure 15.

      Figure 15. An incenter of a triangle with an inscribed circle.

      Median

      A line drawn from a vertex that divides the opposite side of the triangle into two equal segments is called a median. See figure 16

      Figure 16. A median CD.

      The medians of a triangle are concurrent. The point of the medianś intercept is called centroid. The centroid is a geometric center of a triangle. If you cut a triangle out of cardboard, find its centroid and place the triangle on a pencil tip so that the tip is in the centroid of the triangle, the triangle will be perfectly balanced. See figure 17.

      Figure 17. A geometric center of a triangle. AE, CD and BF are medians.

      Perpendicular bisector

      A perpendicular line drawn from a midpoint of a trianglés side is called a perpendicular bisector. See Figure 18.

      Figure 18. Midpoints are: D, H, F; ED _|_AC; JH_|_AB; GF_|_BC; AD = DC; AH = HB; BF=FC;

      Perpendicular bisectors of the sides of a triangle are concurrent. The point, at which the perpendicular bisectors intersect, is called the circumcenter. A circumcenter of a triangle is equidistant from the vertices of the triangle and consequently it is a center of a circumscribed circle. See figure 19.

      Figure 19. O is a center of the circumscribed circle.

      A point that lies on a perpendicular bisector of a triangle side is equidistant from the endpoints of the side. Converse also is true. See figure 20.

      Figure 20. A perpendicular bisector BD.

      If DB is a perpendicular bisector of the side AC then AE = EC and AF=FC.

      In any triangle, the orthocenter, centroid and circumcenter lie on the same line which is called the Euler line named after a Swiss mathematician and physicist Leonhard Euler. The distance from orthocenter O to centroid I is twice as long as the distance from centroid I to circumcenter C. See figure 21.

      Figure 21. Euler Line. (CIO)

      IO=2IC. GF, JH, ED are perpendicular bisectors. Point C is circumcenter.

      AI, BI, KI are medians. Point I is centroid. AO, BO, CO are perpendiculars. Point O is orthocenter.

      Triangle sides

      The sum of the lengths of any two sides of a triangle must be greater than the third side. The difference in length of any two sides of a triangle must be less than the third side. See figure 22.

      Figure 22. Length of a trianglés sides.

      An isosceles triangle

      A triangle with two equal sides is called an isosceles triangle.

      An isosceles triangle has many interesting properties and features:

      1. If two sides of a triangle are equal, then two angles that lie opposite the equal lines are equal. Angle BAD = BCD

      2. If you draw a line from the angle between two equal sides, to the middle point of the third side, that line is a bisector of the angle. See figure 23.

      Figure 23. AB = BC.

      Line BD is a bisector. It divides angle ABC into 2 equal angles ABD and DBC.

      3. A bisector drawn from the angle between equal sides is perpendicular to the third line. It means that angle ADB and angle CDB are straight and equal 90 degrees.

      An altitude drawn from the angle between equal sides is the angle bisector and a median.

      AD = DC.

      The two resulting triangles are equal as well.

      Triangle ABD = triangle DBC.

      An Equilateral triangle

      If all three sides of a triangle are equal, such a triangle is called an equilateral triangle.

      What is true for an isosceles triangle is true for an equilateral triangle as well.

      Additionally, all three angles of an equilateral triangle are an equal 60 degrees. See figure 24.

      Figure 24. An equilateral triangle. Angles A, B and C = 60 degrees.

      AB = BC=AC

      Bisectors drawn from each angle of an equilateral triangle are altitudes and medians at the same time. A median, an altitude or an angle bisector divides an equilateral triangle into two equal triangles with the angles 30, 60 and 90 degrees. See figure 25.

      Figure 25. An equilateral triangle ABC.

      BD is a bisector, altitude and median. Triangles ABD and BDC are equal.

      Pythagorean Theorem

      In right triangles, the square of the hypotenuse BC is equal to the sum of the squares of the legs AB and AC. See figure 26.

      Figure 26. Pythagorean Theorem

      Trigonometric functions: sine, cosine, tangent, cotangent

      Figure 27 Trigonometric functions.

      Sine of an angle A is the ratio of the opposite side to the hypotenuse.

      Cosine of an angle A is the ratio of the adjacent side to the hypotenuse.

      Tangent of an angle A is the ratio of the opposite side to the adjacent side.

      Cotangent of an angle A is the ratio of the adjacent side to the opposite side.