19. Conjugation of two arcs of circles by the third arc

When constructing the conjugation of two arcs of circles by the third arc of a given radius, we can consider three cases: when a conjugating arc of radius R touches given arcs of radii R ₁ and R ₂ from the outside (Fig. 34, a); when it creates an internal touch (fig. 34, b); when internal and external tangencies are combined (Fig. 34, c).

The construction of the center O of the conjugating arc of radius R with an external touch is carried out in the following order: from the center O _{1 with a} radius equal to R + R ₁ , an auxiliary arc is drawn, and from the center O ₂ an auxiliary arc with radius R + R _{2 is} conducted . At the intersection of the arcs, a center O of a conjugate arc of radius R is obtained , and at the intersection of radius R + R ₁ and R + R ₂ with arcs of circles, there are obtained conjugation points A and A ₁ .

The construction of the center O with internal tangency differs in that from the center O _{1 they} carry out an auxiliary circle with a radius equal to R - R ₁ and from the center O _{2 with a} radius R - R ₂ . When combining internal and external tangency from the center O ₁ , an auxiliary circle with a radius equal to R - R ₁ and from the center O ₂ is carried out with a radius equal to R + R ₂ .