AnGeL: Analytic Geometry Lever

Contest geometry problems (e.g., from mathematical olympiads) can be solved using coordinates (a.k.a. analytic geometry). Even challenging geometry problems can often be tackled that way. With the support of algebra (A), we can use coordinates as a lever to solve (lift) a difficult (heavy) geometry (G) problem through a series of manageable steps – just like in the illustration above. But coordinate calculations can be tedious, error-prone, and time-consuming, especially when algebraic expressions grow large. Wouldn't it be great if there were an app designed specifically to handle this kind of work?

We are developing such an app! We aim to launch it in 2026. It is designed to assist:

Students, learning how to solve geometry problems using coordinates
Problem selection committees, assessing the difficulty of solving geometry problems via coordinates
Teachers, coordinators, and team leaders, checking students' coordinate-based solutions

The app will be available at this URL (angel.skmo.sk) and will run directly in the browser–no installation required! You'll be able to use it on various devices (desktops, laptops, tablets, smartphones). It will be completely free of charge and free of ads.

The basic workflow will be:

Introduce points (e.g., A, B, C, I, O, F, G, ...)
Assign coordinates to the points using variables (e.g., B = (b, 0), C = (c, 0), I = (0, r), O = (0, 0), F = (f_x, f_y), G = (g_x, g_y), A = (a_x, a_y), ...)
Specify independent and dependent variables (e.g., b, c, and r are independent; the rest are dependent)
Define the relationships that express how dependent variables are derived from the independent variables and previously computed values (e.g., F is the reflection of O over IC, G is the reflection of O over IB, A is the intersection of BG and CF, ...)
State the claim to be proved (e.g., ∠KIL + ∠YPX = 180°)

Based on the defined relations, AnGeL symbolically calculates the dependent variables in terms of the independent ones (showing intermediate steps) and verifies the validity of the claim.

Want to be the first to know when we launch?
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Once we launch the app, we will send the notification about the launch to the email address filled in this form. We will not use the submitted email address for any other purpose.

This page was prepared by Peter Novotný, Observer A from Slovakia.
The development of the app is funded by the Cultural and Educational Grant Agency (KEGA) of the Ministry of Education, Research, Development and Youth of the Slovak Republic, grant No. 036ŽU-4/2024.