# Equal Incircles Theorem, Angela Drei's Proof

The Equal Incircle Theorem is a generalization of a sangaku problem:

Let C be a point. Assume points M_{i}, I = 1, 2, ..., N (N > 3) lie on a line not through C. Assume further that the incircles of triangles M_{1}CM_{2}, M_{2}CM_{3}, ..., M_{N-1}CM_{N} all have equal radii. Then the same is true of triangles M_{1}CM_{3}, M_{2}CM_{4}, ..., M_{N-2}CM_{N}, and also of triangles M_{1}CM_{4}, M_{2}CM_{5}, ..., M_{N-3}CM_{N}, and so on.

The proof below is by Angela Drei, an Italian mathematics teacher. The proof first appeared in an Italian magazine *Archimede*. It depends on two lemmas.

### Lemma 1

In any ΔABC,

1 - 2r/h = tan(α/2)tan(β/2),

where r is the inradius, h the altitude to AB, α = ∠BAC, β = ∠ABC.

### Proof

Let a, b, c be the lengths sides BC, AC, AB and p = (a + b + c)/2 the semiperimeter of ΔABC. As we know,

tan²(α/2) = (p - b)(p - c) / p(p - a) and tan²(β/2) = (p - a)(p - c) / p(p - b)

Taking the product gives

tan²(α/2)tan²(β/2) = (p - c)² / p².

Next, letting S be the area of ΔABC,

(p - c) / p | = 1 - c/p | |

= 1 - ch·r / pr·h | ||

= 1 - 2Sr / Sh | ||

= 1 - 2r / h, |

which proves the lemma.

### Lemma 2

Given a triangle ABC, D lie on AB, r_{1} and r_{2} are the radii of the circles inscribed in ACD and BCD, r is the inradius, h is the length of the altitude to the side AB. Then the following relation holds:

(1 - 2r_{1}/h)(1 - 2r_{2}/h) = 1 - 2r/h.

In particular, if r_{1} = r_{2}, then (1 - 2r_{1}/h)² = 1 - 2r/h.

### Proof

Apply Lemma 1 to triangles ACD, BCD, and ABC and observe that ∠ADC/2 = 90° - ∠BDC/2 so that

### Lemma 2'

Assume points M_{i}, I = 1, 2, ..., N (N > 3) lie on side AB of ΔABC, _{1},_{N}._{1}CM_{2}, M_{2}CM_{3}, ..., M_{N-1}CM_{N} all have equal radii, say ρ. If, as before, h is the altitude from C and r is the inradius of ΔABC, then

(1 - 2ρ/h)^{N-1} = 1 - 2r/h.

### Proof

Similar to the proof of Lemma 2, apply Lemma 1 to each of the triangles M_{i}CM_{i+1} and multiply the identities. The products of the tangents due to pairs of supplementary angles at points M_{j},

### Proof of the Equal Incircle Theorem

This is a direct consequence of Lemma 2 applied to pairs of adjacent triangles M_{i-1}CM_{i} and M_{i}CM_{i+1}.

## Sangaku

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equal Incircle Theorem, Angela Drei's Proof

- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

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