The inner Quadrilateral of torle's theorem circle

From Wikipedia:

In mathematics, torle's theorem is a quadrilateral Theorem in Euclidean geometry. Torle's theorem indicates that the sum of the product of the two sides of a convex quadrilateral is not less than the product of the two diagonal lines. An equal sign is obtained only when the quadrilateral is a circular inner quadrilateral or degrades to a straight line (also known as Euler's theorem ). The narrow tolemi theorem can also be described as: the product of the two sides of a convex quadrilateral in a circle and the product of the two diagonal lines. Its Inverse Theorem is also true: if the product of the two sides of a convex quadrilateral is equal to the product of the two diagonal lines, the convex quadrilateral is connected to a circle. Torle's theorem can be seen as a method for determining the inner Quadrilateral of a circle.

Geometric proof

1. Set ABCD to an inner quadrilateral.
2. on the string BC, the circumference of the byte BAC = ∠ BDC, and on the AB, the byte ADB = ∠ ACB.
3. Take a little K on the AC so that ∠ ABK = ∠ CBD; Because ∠ ABK + ∠ CBK = ∠ ABC = ∠ CBD + ∠ ABD, ∠ CBK = ∠ ABD.
4. Therefore, △abk is similar to △dbc. Similarly, △abd is similar to △kbc.
5. Therefore, AK/AB = CD/BD and CK/BC = DA/BD;
6. Therefore, AK * BD = AB * CD and CK * BD = BC * DA;
7. The two formula is added to get (AK + CK) * BD = AB * CD + BC * DA;
8. But AK + CK = AC, so AC * BD = AB * CD + BC * DA. Pass.

Geometric proof of Inverse Theorem
Using the ry method, we can also prove the torlemi Theorem and Its Inverse Theorem. Set to any convex quadrilateral. If a triangle is similar to a triangle, the following occurs:

Export ABP = ∠ DBC (Red Corner)
Therefore, mongoabd = mongopbc
At the same time, according to the properties of similar triangles, there are:

AB/BD = BP/BC
It can be seen that the triangle and the triangle are also similar. The two similarities are described as follows:
AB * CD = BD * AP;

AD * BC = BD * PC;
Add the two formulas to get:

AB * CD + AD * BC = (AP + PC) * BD> = AC * DB
An equal sign is equivalent to A, B, C, and D when it is only A, P, and C, and is equivalent to ∠ BAC = ∠ BAP = ∠ BDC. Therefore, the proposition is proved.