This is a special case where the center of the circle is inside the inscribed orange angle. The blue angle is called a central angle.
init({
range: [ [ -RADIUS - 1, RADIUS + 1 ], [ -RADIUS - 1, RADIUS + 1 ] ]
});
addMouseLayer();
graph.circle = new Circle( RADIUS );
style({ stroke: BLUE, fill: BLUE });
graph.circle.drawCenter();
graph.circle.drawPoint( START );
graph.circle.drawPoint( START + MEASURE );
graph.central = graph.circle.drawCentralAngle( START, START + MEASURE );
style({ stroke: ORANGE, fill: ORANGE });
graph.inscribed = graph.circle.drawInscribedAngle( INSCRIBED, START, START + MEASURE );
graph.circle.drawMovablePoint( INSCRIBED, START + MEASURE, START );
What do we know about the sub-angles formed by the dashed diameter shown above?
style({stroke: BLUE, "stroke-dasharray": "-"}, function() {
graph.circle.drawChord( INSCRIBED, INSCRIBED + 180 );
});
If we only look at the sub-angles drawn now, we see that this is the special case from the previous inscribed angles exercise!
We know that the orange sub-angle is one half the measure of the blue sub-angle.
graph.inscribed.arc.animate({opacity: 0.4});
graph.central.arc.animate({opacity: 0.4});
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( INSCRIBED + 180, START + MEASURE, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( INSCRIBED, INSCRIBED + 180, START + MEASURE, 0.7 );
Likewise, the other orange sub-angle is one half the measure of the other blue sub-angle, as shown.
graph.centralSub.remove();
graph.inscribedSub.remove();
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( START, INSCRIBED + 180, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( INSCRIBED, START, INSCRIBED + 180, 0.7 );
If both orange sub-angles are one half both blue sub-angles, then we know that the original orange angle is one half the original blue angle.
graph.centralSub.remove();
graph.inscribedSub.remove();
graph.inscribed.arc.animate({opacity: 1.0});
graph.central.arc.animate({opacity: 1.0});
\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{\text{blue angle}}
\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{MEASURE^{\circ}}
\color{ORANGE}{\text{orange angle}} = \color{BLUE}{MEASURE / 2^{\circ}}