The minimal triangle circumscribing a semicircle

Consider a fixed semicircle with center at O and radius r. Let \triangle ABC be the isosceles triangle which circumscribes it (Figure 1).

triangle_circumscribing_semicircleFigure 1

What will be the characteristics of the minimal form of the said triangle, i.e. triangle with minimum perimeter, which will circumscribe the said semicircle? Let the base angles of the isosceles triangle be \angle x. In order to determine the equation for the perimeter with respect to this base angle this we perform the following construction (Figure 2):
1) Join point O to one of the points of tangency of the circumscribing triangle ABC and the semi-circle (Point I_1, I_2).
2) This gives us two right triangles \triangle OI_2C and \triangle AI_2O.
3) Using their sides and \angle x we can determine the lengths of the segments \overline{OC}, \overline{I_2C}, \overline{I_2A} (Figure 2). Twice their sum will yield the equation of the perimeter of the triangle. It is:

P=2r\left (\tan(x)+ \dfrac{1}{\tan(x)}+\dfrac{1}{\sin(x)}\right)

triangle_circumscribing_semicircle2Figure 2

Thus, we will get the minimal triangle when the function y=\left (\tan(x)+ \tfrac{1}{\tan(x)}+\tfrac{1}{\sin(x)}\right) attains a minimum (Figure 3).

semicircle_triangle_minimFigure 3

The derivative of this function is (Figure 3):
y'=\left (\dfrac{1}{\cos^2(x)}- \dfrac{1}{\sin^2(x)}-\dfrac{\cos(x)}{\sin^2(x)}\right)

We get our minimum when y'=0. Thus at minimum,

\dfrac{1}{\cos^2(x)}=\dfrac{1}{\sin^2(x)}+\dfrac{\cos(x)}{\sin^2(x)}\\[7pt] \sin^2(x)=\cos^3(x)+\cos^2(x)\\[7pt] 1-\cos^2(x)=\cos^3(x)+\cos^2(x)\\[7pt] \cos^3(x)+2\cos^2(x)-1=0

Putting y=\cos(x) we get the cubic equation y^3+2y-1=0. It can be factorized as (y + 1) (y^2 + y - 1) = 0. From which we get the roots as (-\phi, -1, \phi'), where \phi=\tfrac{1+\sqrt{5}}{2} is the Golden ratio and \phi'=\tfrac{\sqrt{5}-1}{2}=\tfrac{1}{\phi}. x=\arccos(y). Since x is the base angle of an isosceles triangle, y=\phi' is the only valid value for which the \arccos(y) may be taken. Thus we get,

x=\arccos(\phi') \approx 0.9045569

Thus, when the base angles of the isosceles \triangle ABC are \arccos(\phi') then we have the minimal circumscribing triangle of a given semicircle (Figure 4).

triangle_circumscribing_semicircle_shrIFigure 4

The special features of this triangle can be seen in Figure 4, where the radius of the fixed semicircle is taken to be a unit: It is made up 2 right triangles whose sides specify the equation of \phi, i.e. \phi^2-\phi-1=0. The various powers of \phi are captured by it. The perimeter of the minimal circumscribing triangle is thus 2r \phi^{5/2}. The 2 points where the triangle touches the circular arc of the semicircle are specified by the angle \arcsin(\phi').

This triangle has a special place in the great geometric manifestation of the Bhairava and the Bhairavī, known as the Śrī-yantra, versions of which are worshiped in the Paścimāṃnāya (Kubjikā; Western lodge) and Dakṣiṇāṃnāya (Śrī-kula; Southern lodge) of the kaula tradition. This yantra is described in a key text of these traditions, the Saundaryalaharī, thusly:
caturbhiḥ śrīkaṇṭhaiḥ śivayuvatibhiḥ pañcabhir api
prabhinnābhiḥ śaṃbhor navabhir api mūlaprakṛtibhiḥ ।
tryaś-catvāriṃśad vasu-dala-kalāśra-trivalaya
tri-rekhābhiḥ sārdhaṃ tava śaraṇa-koṇāḥ pariṇatāḥ ॥

shrI_yantraFigure 5

The above verse from the Saundaryalaharī may be rendered as:
The (tri)angles in what constitute your dwelling, which is built of 9 mūla-prakṛti-s, i.e. the 4 Śrīkaṇṭha-s and the 5 Śiva-yuvati-s, all [stationed] apart from Śambhu, along with the lotuses with Vasu- (8) and lunar-digit- (16) petals, the 3 circles and the 3 lines, are 43 in number. (modified version of the translation of Subrahmanya Shastri and Shrinivasa Ayyangar)

This verse precisely captures the essence of the structure of the Śrī-yantra. The 9 mūla-prakṛti-s are the 9 basic triangles that constitute to the yantra. They represent the 9-fold form of the Bhairava of the Paścimāṃnāya, namely Navātman-bhairava. They are in turn divided into: 4 upward-facing or liṅga triangles, which are termed Śrīkaṇṭha-s. These are the manifest forms of the Bhairava. 5 downward-facing or yoni triangles, which are termed Śiva-yuvati-s. These are the manifest forms the Bhairavī. The central point, the bindu, lies apart from from them and represents the Bhairava in his state as the unmanifest singularity, i.e. Akulavīra. In contrast, the rest of the yantra with is manifold geometric structures can be seen as his śakti, Kaulinī, who manifests diversely as the kula-s of Yoginī-s. These Yoginī-s inhabit all elements of the structure: For example, the 8 mātṛkā-s (Caṇḍikā, Brāhmī, Rudrāṇī, Kaumārī, Vaiṣṇavī, Vārāhī, Indrāṇī, Cāmuṇḍā) inhabit the 8-petalled lotus (This is as per the mūla-prakrīyā of the Ānanda-bhairava tradition. Other paramparā-s place the Kāma-yoginī-s from Anaṅgapuṣpā to Anaṅgamālinī in these petals). The 16 Śrī-nityā-s inhabit the 16-petalled lotus.

The 43 triangles are reckoned as the 42 outward-pointing triangles and the inner-most triangle formed by the intersection of the 9 basic triangles. The outermost of these has 14 triangles known as the Saubhāgyadāyaka in which dwell the Sarva-saubhāgyadāyaka-yoginī-s. The second rung with 10 triangles is known as the Sarvārthasādhaka with eponymous yoginī-s. The third rung again with 10 triangles is the Sarva-rakṣākara with the 10 yoginī-s of that kula. The 4th rung with 8 triangles is the Sarva-rogahara with the 8 Vāgdevī-s as its resident yoginī-s. The 5th rung with the central triangle around the bindu has Kāmeśvarī as its yoginī.

So, where does our above minimal triangle figure in all this? It defines the Śrīkanṭha-1 which is the first triangle to be drawn while constructing the Śrī-yantra. Also, notably, the first Śiva-yuvati is an isosceles triangle formed by welding two 3-4-5 right triangles along the side of length 4. As we have seen before, this 3-4-5 triangle, the most primitive of the integral right triangles, emerges naturally as the maximal triangle defined by the characteristic ellipse of a right triangle. Thus, the two primary triangles of the Śrī-cakra natural feature as the ultima of certain problems of triangle geometry.

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