PROPOSITION XVII. THEOREM.

807. Two symmetrical spherical triangles are equivalent.

B

Let ABC, A'B'C' be two symmetrical spherical triangles with their homologous vertices opposite each to each.

To prove that the triangles ABC, A'B'C' are equivalent. Proof. Let P be the pole of a small circle passing through the points A, B, C, and let POP' be a diameter.

Draw the great circle arcs PA, PB, PC, P'A', P'B', P'C'.

.. the two symmetrical ▲ PAC and P'A'C' are isosceles.

and

Ax. 9

▲ A'B'C' ▲ P'A'C' + ^ P'A'B' + ^ P'B'C'.

.. A ABC≈ ▲ A'B'C'.

Q. E. D.

If the pole P should fall without the ▲ ABC, then P' would fall without ▲ A'B'C', and each triangle would be equivalent to the sum of two symmetrical isosceles triangles diminished. by the third; so that the result would be the same as before.

PROPOSITION XVIII. THEOREM.

808. Two triangles on the same sphere or equal spheres are equal, if two sides and the included angle, or two angles and the included side, of the one are respectively equal to the corresponding parts of the other and arranged in the same order.

PROPOSITION XIX. THEOREM.

809. Two triangles on the same sphere or equal spheres are symmetrical, if two sides and the included angle, or two angles and the included side, of the one are equal, respectively, to the corresponding parts of the other and arranged in the reverse order.

Then A ABC can be superposed upon the A DEF", so that they will coincide as in the corresponding case of plane A.

But A DEF and DEF are symmetrical by construction. .. AABC, which coincides with A DEF", is symmetrical with respect to ▲ DEF.

PROPOSITION XX. THEOREM./

Q.E. D.

810. Two mutually equilateral triangles on the same sphere or equal spheres are mutually equiangular, and are equal or symmetrical.

§ 237 § 583

Therefore, the corresponding dihedral are equal. Hence, the of the spherical A are respectively equal. Therefore, the A are equal or symmetrical, according as their equal sides are arranged in the same or reverse order.

Q. E. D.

Ex. 737. The radius of a sphere is 4 inches. From any point on the surface as a pole a circle is described upon the sphere with an opening of the compasses equal to 3 inches. Find the area of this circle.

Ex. 738. The edge of a regular tetrahedron is a. Find the radii R, R' of the inscribed and circumscribed spheres.

Ex. 739. Find the diameter of the section of a sphere 10 inches in diameter made by a plane 3 inches from the centre.

811. Two mutually equiangular triangles on the same sphere or equal spheres are mutually equilateral, and are either equal or symmetrical.

Let the spherical triangles T and T' be mutually equiangular. To prove that T and T' are mutually equilateral, and equal or symmetrical.

Proof. Let the AP be the polar ▲ of T, and P' of T'. By hypothesis, the ▲ T and T' are mutually equiangular. .. the polar AP and P' are mutually equilateral. ..the polar AP and P' are mutually equiangular. But the AT and T' are the polar A of P and P'. .. the AT and T" are mutually equilateral. Hence, the AT and T" are equal or symmetrical.

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§ 810

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§ 793

§ 810

Q.E. D.

NOTE. The statement that mutually equiangular spherical triangles are mutually equilateral, and equal or symmetrical, is true only when they are on the same sphere, or equal spheres. But when the spheres are unequal, the spherical triangles are unequal; and the ratio of their homologous sides is equal to the ratio of the radii of the spheres.

§ 465

Ex. 740. At a given point in a given arc of a great circle, to construct

Ex. 741. To inscribe a circle in a given spherical triangle.

PROPOSITION XXII. THEOREM.

812. In an isosceles spherical triangle, the angles opposite the equal sides are equal.

In the spherical triangle ABC, let AB equal AC.

B

D

Proof. Draw the arc AD of a great circle, from the vertex A to the middle of the base BC.

Then A ABD and ACD are mutually equilateral.

.. A ABD and ACD are mutually equiangular. § 810

813. COR. The arc of a great circle drawn from the vertex of an isosceles spherical triangle to the middle of the base bisects the vertical angle, is perpendicular to the base, and divides the triangle into two symmetrical triangles.

Ex. 742. To circumscribe a circle about a given spherical triangle. Ex. 743. Given a spherical triangle whose sides are 60°, 80°, and 100°. Find the angles of its polar triangle.

Ex. 744. Given a spherical triangle whose angles are 70°, 75°, and 95°. Find the sides of its polar triangle.

Ex. 745. Find the ratio of two homologous sides of two mutually equiangular triangles on spheres whose radii are 12 inches and 20 inches.