Kepler triangle

A Kepler triangle is a special right triangle with edge lengths in geometric progression. The progression can be written: $1:\sqrt {\varphi }:\varphi,$ or approximately $1:1.272:1.618.$ When an isosceles triangle is formed from two Kepler triangles, reflected across their long sides, it has the maximum possible inradius among all isosceles triangles having legs of a given size. Most of the properties described below were discovered by the famous Russian mathematician Lev Emelianov, who in his works called this isosceles triangle ”aureate triangle”.

Sides and angles of doubled Kepler triangle

Let’s define the doubled Kepler triangle as triangle which has the maximum possible inradius among all isosceles triangles having legs of a given size.

Let the incircle of an isosceles $\triangle ABC (AB = AC)$ touch the sides $AB$ and $BC$ at points $K$ and $M, KI = MI = r,$ $\angle BAM = \alpha, \angle ABC = 2 \beta, \alpha + 2 \beta = 90^\circ.$ We need to find minimum of $\frac {AB}{r} = \cot \alpha +\cot \beta.$ Let us differentiate this function with respect $\beta$ to taking into account that $0<\alpha,2 \beta < 90^\circ, \frac {d \alpha}{d \beta} = -2: \frac {-2}{\sin^2 \alpha}+\frac {1}{\sin^2 \beta} = 0 \implies$ $\sqrt{2} \sin \beta = \sin \alpha = \cos 2 \beta = 1 - 2 \sin^2 \beta \implies \sin \alpha = 1 - \sin^2 \alpha.$ Therefore $\sin \alpha = \cos 2 \beta = \frac {\sqrt{5} - 1}{2} = \phi = \frac{1}{\varphi}.$

Let $AB = 1 \implies BM = BK = \phi, AM = \sqrt{\phi}, AK = \phi^2,$ $AI = \phi \cdot \sqrt{\phi}, BI = \sqrt{2} AI, IK = IM = \phi^2 \cdot \sqrt{\phi}.$ vladimir.shelomovskii@gmail.com, vvsss

Construction of a Kepler triangle

Let $M$ be the midpoint of the base $BC,$ $DM \perp BC, DM = BC.$ Point $E \in BD, DE = BC.$

The point $F$ is the intersection of a circle with diameter $BM$ and a circle centered at point $B$ and radius $BE$ , which is located in the half-plane $BC$ where there is no point $D$ .

The bisector of the obtuse angle between lines $BF$ and $BC$ intersects bisector $BC$ at the vertex $A$ of the Kepler triangle.

The construction is based on the fact that $\cos 2 \angle ABC = 2 - \sqrt{5}.$

Properties of a Kepler triangle

Let $I,H,O,$ and $F$ be the incenter, ortocenter, circumcenter, and Feuerbach point (midpoint $BC),$ respectively. $\phi = \frac {\sqrt{5} - 1}{2} = \frac {1}{\varphi}, \phi^2 + \phi = 1, AB = 1.$

The remaining notations are shown in the figure.

Prove:

1. Points $E,F',E'$ are collinear.

2. $AH = IF.$

3. Points $D,H,D'$ are collinear.

4. $AF' = EH = IH.$

5. $IO = FO.$

6. $P = CH \cap ID' \implies PI = PH = PD'.$

7. $2 BO \cdot AF = AB^2.$

Proof

It is known that $BF = BD = \phi, AF = \sqrt{\phi}, AD = \phi^2, AI = \phi \cdot \sqrt{\phi}, r=ID = IF = IF' = \phi^2 \cdot \sqrt{\phi},$ $\sin \alpha = \phi, \cos \alpha = \sqrt{\phi}\implies \sin 2\alpha = 2 \phi \sqrt{\phi}, \cos 2\alpha = \phi^3.$ 1. $BE = BC \sin \alpha = 2 \phi \cdot \phi = 2 \phi^2.$ Distance from $E$ to $BC$ is $BE \cos \alpha = 2 \phi^2 \cdot \sqrt{\phi} = 2r.$

2. $AH = \frac {AE}{\cos \alpha} = \frac{1-BE}{\sqrt{\phi}}= \frac{\phi - \phi^2}{\sqrt{\phi}}= \phi^2 \cdot \sqrt{\phi} = r.$

3. Distance from $D$ to $BC$ is $BD \cos \alpha = \phi \cdot \sqrt{\phi} = AI = HF.$

4. $ED = EF', \angle DHE = \angle EAF' = \alpha \implies AF' = HE.$ $EH = DH \cos \alpha = DH \tan \alpha = HI.$ 5. $\angle BOF = 2 \alpha \implies FO = BF \cot 2 \alpha = \phi \cdot \frac{\phi^3}{ 2 \phi \sqrt{\phi}} = \frac {r}{2}.$

6. $\angle CHD' = \angle ID'H \implies PI = PH = PD', AH = IF \implies P \in$ midline of $\triangle ABC.$

7. $R = BO = \frac{BF}{\sin 2 \alpha} = \frac{1}{2\sqrt{\phi}} \implies 2R \cdot AF = AB^2.$

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Kepler triangle

Sides and angles of doubled Kepler triangle

Construction of a Kepler triangle

Properties of a Kepler triangle