Most students are already familiar with the three standard triangle congruence rules taught in school: SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle). These form the foundation of geometry and are commonly seen in classroom exercises and exams, including the GCSE Mathematics syllabus.

But what happens when we encounter problems that don’t follow these standard patterns directly? Can we still prove triangle congruence using medians, altitudes, angle bisectors, or partial side information?

In this section, we explore a selection of challenging triangle congruence problems that require deeper reasoning and clever constructions. These problems go beyond the textbook basics and are perfect for students preparing for Olympiad-style questions, GCSE extensions, or anyone wanting to sharpen their proof skills.

Problems

Prove the criterion for triangle congruence by two angles and perimeter.
Prove the criterion for triangle congruence by the median and the two angles that the median divides at the vertex.
Prove the criterion for triangle congruence by a, ma, and ha.
In triangle ABC, the angle bisector of ∠A meets side BC at point D. Points E and F lie on sides AB and AC, respectively, such that ∠EBD = ∠FCD. Prove that triangles EBD and FCD are congruent.
In triangles ABC and A′B′C′, suppose AB = A′B′, the altitudes from C and C′ to AB and A′B′ are equal, and ∠C = ∠C′. Prove that the triangles are congruent.
Let triangles ABC and A′B′C′ have equal medians from A and A′, and AB = A′B′, AC = A′C′. Prove that the triangles are congruent.
Triangle ABC is reflected over the angle bisector of ∠A. Prove that the reflected triangle is congruent to the original triangle, and identify which parts coincide.
On the extension of the base AC of an isosceles triangle ABC beyond point C, choose an arbitrary point D. On segment BD, mark points K and L such that ∠BDA = ∠KCD = ∠LAD. Prove that triangles BCK and ABL are congruent.
Using the previous problem, prove the criterion for triangle congruence by a, α (alpha) and b+c.

Challenging Triangle Congruence

Properties of a Right-Angled Triangle

Circumcircle of a Triangle

Inscribed Quadrilateral

Geometry Warm-Up: Angles and Lines

Axioms and Postulates of Euclid

Morley's Triangle

Minimum and Maximum Problems in Geometry

Where do you hold your classes?

We hold our classes online or on-site on Saturdays at our branch in Pimlico Academy, London.
You can find our timetable here.

What do you need to start learning online?

For lessons you only need a computer or phone with a microphone, camera and Internet access. Wherever you are - in London, Nottingham, New York or Bali - online lessons will be at hand.

When can I take the trial lesson?

You can get acquainted with the school at any time convenient for you. To do this, just leave a request and sign up for a lesson.

What should I expect from the trial lesson?

The trial lesson is a 30-minute online session designed to get a sense of how your child approaches mathematical thinking and problem solving. (In practice, it often runs a bit longer if the student is engaged!)

We typically explore a range of fun and challenging problems drawn from competitions. We adapt the difficulty based on how the student responds, aiming to make it both accessible and stimulating.

After the session, we’ll have a quick conversation with the parent to share observations and suggest a personalised path forward.

I can't attend class, what should I do?

It is OK, it happens! Students have the opportunity to cancel a lesson up to 8 hours before the scheduled time without loss of payment. So you can reschedule it for a convenient time, and the teacher will have the opportunity to

I don't have much free time, will I have time to study?

Learning can take place at your own pace. We will select a convenient schedule and at any time we will help you change the schedule, take a break or adjust the program.

How long is one lesson?

All classes last 1 hour.

Meet our team

Our teachers will tell you how to prepare for exams, help you cope with difficult tasks and win the Olympiad
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They will tell you about the pitfalls of exams and the most common mistakes, and explain how to avoid them

George Ionitsa

Founder &
Maths and Coding Coach

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