mathematics

Mixtilinear incircles of a triangle

Mixtilinear incircles of a triangle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mixtilinear incircles of a triangle rather than just read about it. In short: In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex A {\displaystyle A} is called the A {\displaystyle A} -mixtilinear incircle.

Key takeaways

Mixtilinear incircles of a triangle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mixtilinear incircles of a triangle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mixtilinear incircles of a triangle from memory before moving on to harder problems.

Reference excerpt

In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex A {\displaystyle A} is called the A {\displaystyle A} -mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.

Proof of existence and uniqueness The A {\displaystyle A} -excircle of triangle A B C {\displaystyle ABC} is unique. Let Φ {\displaystyle \Phi } be a transformation defined by the composition of an inversion centered at A {\displaystyle A} with radius A B ⋅ A C {\displaystyle {\sqrt {AB\cdot AC}}} and a reflection with respect to the angle bisector on A {\displaystyle A} . Since inversion and reflection are bijective and preserve touching points, then Φ {\displaystyle \Phi } does as well. Then, the image of the A {\displaystyle A} -excircle under Φ {\displaystyle \Phi } is a circle internally tangent to sides A B , A C {\displaystyle AB,AC} and the circumcircle of A B C {\displaystyle ABC} , that is, the A {\displaystyle A} -mixtilinear incircle. Therefore, the A {\displaystyle A} -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to B {\displaystyle B} and C {\displaystyle C} .

Construction

The A {\displaystyle A} -mixtilinear incircle can be constructed with the following sequence of steps.

Draw the incenter I {\displaystyle I} by intersecting angle bisectors. Draw a line through I {\displaystyle I} perpendicular to the line A I {\displaystyle AI} , touching lines A B {\displaystyle AB} and A C {\displaystyle AC} at points D {\displaystyle D} and E {\displaystyle E} respectively. These are the tangent points of the mixtilinear circle. Draw perpendiculars to A B {\displaystyle AB} and A C {\displaystyle AC} through points D {\displaystyle D} and E {\displaystyle E} respectively and intersect them in O A {\displaystyle O_{A}} . O A {\displaystyle O_{A}} is the center of the circle, so a circle with center O A {\displaystyle O_{A}} and radius O A E {\displaystyle O_{A}E} is the mixtilinear incircle This construction is possible because of the following fact:

Verrier's lemma The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Mixtilinear incircles of a triangle

Start with the simplest possible case. Write down what Mixtilinear incircles of a triangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mixtilinear incircles of a triangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mixtilinear incircles of a triangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mixtilinear incircles of a triangle

In research: Mixtilinear incircles of a triangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mixtilinear incircles of a triangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mixtilinear incircles of a triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean plane geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Mixtilinear incircles of a triangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Mixtilinear incircles of a triangle” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Mixtilinear incircles of a triangle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mixtilinear incircles of a triangle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mixtilinear incircles of a triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mixtilinear incircles of a triangle in simple terms?

Why does Mixtilinear incircles of a triangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mixtilinear incircles of a triangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mixtilinear incircles of a triangle.