# Circumcircle of a square and an arbitrary point inside it; prove: $|A_1B_1|\cdot|C_1D_1|=|A_1D_1|\cdot|B_1C_1|$

Point $$T$$ is inside the square $$ABCD$$. Let $$A_1,B_1,C_1,D_1$$ the other intersection point of the lines $$AT,BT,CT,DT$$ respectively and the circumcircle of the square $$ABCD$$. Prove: $$|A_1B_1|\cdot|C_1D_1|=|A_1D_1|\cdot|B_1C_1|$$

My attempt:

I was looking for the inscribed angles of the same measure: $$\measuredangle ABB_1=\measuredangle AA_1B_1\;\&\;\measuredangle BTA=\measuredangle B_1TA_1\implies\;\Delta ABT\;{\sim}\;\Delta A_1B_1T$$ Analogously:

$$\Delta TAD_1{\sim}\Delta TA_1D$$$$\;\Delta C_1D_1T\;{\sim}\Delta CDT$$$$\Delta D_1A_1T{\sim}\Delta DAT$$$$\Delta B_1C_1T{\sim}\Delta CBT$$

Also, $$\measuredangle DB_1B=\measuredangle BA_1D$$, so $$DB_1B$$ and $$BA_1D$$ are right triangles.

However, I couldn't find any triangles with useful information. May I ask for advice on solving the problem? Thank you in advance!

Because $$\frac{A_1D_1\cdot B_1C_1}{A_1B_1\cdot C_1D_1}=\frac{\frac{A_1D_1}{AD}\cdot\frac{B_1C_1}{BC}}{\frac{A_1B_1}{AB}\cdot\frac{C_1D_1}{CD}}=\frac{\frac{A_1T}{DT}\cdot\frac{C_1T}{BT}}{\frac{A_1T}{BT}\cdot\frac{C_1T}{DT}}=1$$