Three Centered Arch II: Geometric Construction
A three-centered arch consists of three segments of circles smoothly "glued" together. In this Demonstration you can choose to show a black line on the left axis to mark the width of the arch using the checkbox "left". You can mark and move a target height of the arch using the blue segment ("height"). Then you can select different positions of the red, green, and blue locators to reach this marked height. When you select approximately correct parameters, little red circles appear at the top of the arch (when the arch touches the blue marker at the top) and also on the right part of the axis (when the two green lines there overlap). If the tangents are identical, then the segments of the circles join smoothly.
This construction uses two theorems of Euclid: "If two circles touch one another their centers and the point of contact lie on a straight line (III, 11 and III 12)". "The straight line from the center of a circle to the point of contact of a tangent and the tangent are perpendicular (III, 18)".
In practice the width is often divided in three parts. Three centered arches are used as door or window arches and also for arches of bridges. The construction is similar to the construction of ovals. There also exists a construction with "a string and three nails".
There are similar constructions for arches with four and five centers. The four-centered arch has a vertex at the highest point.