Similar triangles: Preliminary Theorem: A straight line on one side of a parallel triangle intersects with the other two sides (or extension lines on both sides) to form a triangle similar to the original triangle.

Decision Theorem 1: If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar.

Simply put, two angles are equal and two triangles are similar.

Decision Theorem 2: If two sides of a triangle are in direct proportion to two sides of another triangle, and the included angles are equal, then the two triangles are similar.

To put it simply: the two sides are proportional and the included angle is equal, and the two triangles are similar.

Decision Theorem 3: If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar.

It can be simply said that three sides are proportional and two triangles are similar.

(1) The similarity between two equal right triangles corresponds to an acute angle.

(2) If the two right-angled sides of two right-angled triangles are proportional, then the two triangles are similar.

Theorem for judging the similarity of right-angled triangles: If the hypotenuse and a right-angled side of one right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.

Similar triangles's property theorem: the ratio of similar triangles to height, to the median line and to the angular bisector are all equal to similarity ratio.

Similar triangles area ratio and circumscribed circle area ratio are equal to the square of similarity ratio.

Theorem of intersecting chords: The product of two intersecting chords in a circle is equal. Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the length ratio of the two lines where this point intersects the secant. Secant theorem: from a point outside the circle p, two secants intersect the circle at a and b respectively; C, d, with pa Pb = PC PD. Unified induction: If two straight lines L 1 and L2 are drawn through the point P that is not on the circle, L 1 and the circle intersect at A and B (which can be coincident, that is, tangent), and L2 and the circle intersect at C and D (which can be coincident), there is Pa Pb = PC PD.