AMSCO'S Geometry. New York - Rye High School

528 Ratio, Proportion, and SimilarityTheorems12.1 In a proportion, the product of the means is equal to the product of theextremes.12.1a In a proportion, the means may be interchanged.12.1b In a proportion, the extremes may be interchanged.12.1c If the products of two pairs of factors are equal, the factors of one paircan be the means and the factors of the other the extremes of a proportion.12.2 A line segment joining the midpoints of two sides of a triangle is parallelto the third side and its length is one-half the length of the third side.12.3 Two line segments are divided proportionally if and only if the ratio ofthe length of a part of one segment to the length of the whole is equal tothe ratio of the corresponding lengths of the other segment.12.4 Two triangles are similar if two angles of one triangle are congruent totwo corresponding angles of the other. (AA)12.5 Two triangles are similar if the three ratios of corresponding sides areequal. (SSS)12.6 Two triangles are similar if the ratios of two pairs of corresponding sidesare equal and the corresponding angles included between these sides arecongruent. (SAS)12.7 A line is parallel to one side of a triangle and intersects the other twosides if and only if the points of intersection divide the sides proportionally.12.8 Under a dilation, angle measure is preserved.12.9 Under a dilation, midpoint is preserved.12.10 Under a dilation, collinearity is preserved.12.11 If two triangles are similar, the lengths of corresponding altitudes havethe same ratio as the lengths of any two corresponding sides.12.12 If two triangles are similar, the lengths of corresponding medians havethe same ratio as the lengths of any two corresponding sides.12.13 If two triangles are similar, the lengths of corresponding angle bisectorshave the same ratio as the lengths of any two corresponding sides.12.14 Any two medians of a triangle intersect in a point that divides eachmedian in the ratio 2 : 1.12.15 The medians of a triangle are concurrent.12.16 The altitude to the hypotenuse of a right triangle divides the triangleinto two triangles that are similar to each other and to the original triangle.12.16a The length of each leg of a right triangle is the mean proportionalbetween the length of the projection of that leg on the hypotenuse andthe length of the hypotenuse.12.16b The length of the altitude to the hypotenuse of a right triangle is themean proportional between the lengths of the projections of the legs onthe hypotenuse.12.17 A triangle is a right triangle if and only if the square of the length of thelongest side is equal to the sum of the squares of the lengths of the othertwo sides.