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Magazine Problem 1
Show that \(\angle A O B\) is positive if and only if \(\angle B O A\) is negative.
Problem 3
Exercise. Assume that the angles \(A B C\) and \(A^{\prime} B^{\prime} C^{\prime}\) have the same sign and $$ 2 \cdot \angle A B C \equiv 2 \cdot \angle A^{\prime} B^{\prime} C^{\prime} $$ Show that \(\angle A B C=\angle A^{\prime} B^{\prime} C^{\prime}\).
Problem 4
Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function. Assume \(f(a)\) and \(f(b)\) have opposite signs, then \(f\left(t_{0}\right)=0\) for some \(t_{0} \in[a, b] .\) The intermediate value theorem is assumed to be known; it should be covered in any calculus course. We will use only the following corollary:
Problem 13
Show that two points \(X, Y \notin\) \(\notin(P Q)\) lie on the same side of \((P Q)\) if and only if the angles \(P X Q\) and \(P Y Q\) have the same sign.
Problem 14
Let \(\triangle A B C\) be a nondegenerate triangle, \(A^{\prime} \in[B C]\) and \(B^{\prime} \in[A C] .\) Show that the segments \(\left[A A^{\prime}\right]\) and \(\left[B B^{\prime}\right]\) intersect.
Problem 18
Given \(r>0\) and \(\varepsilon>0,\) there is \(\delta>0\) such that $$ 0<\beta<\delta \quad \Longrightarrow \quad s(r, \beta)<\varepsilon $$ Proof. Fix two points \(A\) and \(B\) such that \(A B=r\). Choose a point \(X\) such that \(\angle A B X\) is positive. Let \(Y \in[A X)\) be the point such that \(A Y=\frac{\varepsilon}{2} ;\) it exists by Proposition 2.4 Note that \(X\) and \(Y\) lie on the same side of \((A B) ;\) therefore, \(\angle A B Y\) is positive. \(\quad\) Set \(\delta=\) \(=\angle A B Y\) Suppose \(0<\beta<\delta\); by Axiom IIIa, we can choose \(B\) so that \(\angle A B C=\beta\) and \(B C=r\). Further we can choose a half-line \([B Z)\) such that \(\angle A B Z=\frac{1}{2} \cdot \beta\). Note that \(A\) and \(Y\) lie on opposite sides of \((B Z)\) and moreover \(\angle A B Z \equiv\) \(\equiv-\angle C B Z .\) In particular, \((B Z)\) intersects \([A Y] ;\) denote by \(D\) the point of intersection. Since \(D\) lies between \(A\) and \(Y\), we have that \(A D
Problem 20
Show that two circles intersect if and only if $$ |R-r| \leqslant d \leqslant R+r $$ where \(R\) and \(r\) denote their radiuses, and \(d\) \- the distance between their centers.
