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Proof.

AD = ha
BE = my.

AC=b.
AD is an altitude as Z ADC = rt. Z.

CE= EA.

.:. EB is a median. 243. The following rules express the procedure in a general form :

(1) Construct a diagram as if the construction were completed.

(2) Find all known lines and angles of the figure and mark them).

(3) Try to construct some part of the figure, usually a triangle.

(4) Make this triangle (or other figure) the basis of the construction.

(4 a) If it is impossible to find such a triangle, draw additional lines and proceed as before.

Ex. 409. To construct a triangle, having given the median and altitude to one side and another median.

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Analysis. (1) Suppose ABC is the required triangle.

(2) Then we know BD (=hy), BE (= mx), CF (= m.), EH(= {m), BH(= jm), HF(= jm.), HC(= j m.), and ABDE and BDA=rt. 4).

(3) Examine all triangles. BED is the only one that can be constructed.

(4) Make the A BED the basis of the construction.

Some problems require the drawing of additional lines (as stated above under 4a).

244. REMARK. When a sum or a difference is given, construct such sum or difference in the analysis.

Ex. 410. To construct a triangle, having given the base, the sum of the other two sides, and the angle included by the two.

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Analysis. (1) Suppose BCD is the required triangle. (2) Then we know two parts only.

.. produce CD to E so that DE= DB. Draw EB. We now know CB(= a), CE(=b+c), 2 CDB (= 4), ZE

and < DBE =

2 (3) Examine all triangles. As BCE can be constructed, make this the basis of the construction.

(=4)

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Ex. 411. To construct a trapezoid, having given the sum of the bases (s), a diagonal (d), a base angle (A), and the angle formed by the diagonals (O).

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Analysis. (1) Suppose ABCD is the required trapezoid. Produce AD to E so that DE= BC. Draw CE.

(2) We know AE(=s), AC(=d), LAOD(=O), ZBAD(=A), LACE(= AOD= 0).

(3) A ACE can be constructed.

Ex. 412. Construct an isosceles triangle, having given the base and the vertical angle.

Ex. 413. Construct an isosceles triangle, having given the sum of base and an arm, and a base angle.

Ex. 414. Construct a triangle, having given an angle, an adjacent side, and the difference of the other two sides.

Ex. 415. Construct a triangle, having given the base, the difference of the other two sides, and the angle included by the two.

Construct a right triangle, having given :
Ex. 416. One arm and the altitude upon the hypotenuse.
Ex. 417. The hypotenuse and the difference between the arms.
Ex. 418. The hypotenuse and the sum of the arms.

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245. A locus of a point in a plane is a line or a group of lines, all points of which fulfil a certain condition, fulfilled by no other points.

246. All points equidistant from two given points lie in the perpendicular-bisector of the line joining the points, and there is no point without the perpendicular-bisector which is equidistant from these points.

... The perpendicular-bisector is a locus.

247. To prove therefore that a certain line is a locus, we must establish that:

(1) Every point in the line satisfies the given condition.
(2) No point without the line satisfies the given condition.

THEOREMS

248. The locus of a point equidistant from the ends of a given line is the perpendicular-bisector of that line.

249. The locus of a point at a given distance from a given point is the circumference described from the point with the given distance as radius.

250. The locus of a point that is at a given distance from a given straight line consists of two lines parallel to the given line at the given distance.

251. The locus of a point equidistant from two given parallel lines is a third parallel, bisecting any line ending in the given parallels.

252. The locus of a point equidistant from two intersecting straight lines, consists of the bisectors of the included angles.

Ex. 429. Find the locus of the vertex of all right angles whose sides pass through two given points.

Ex. 430. Find the locus of the midpoints of the radii of a given circle.

EXERCISES

In the following exercises, state under what conditions no point, one point, or several points may be found.

Ex. 431. In a given line, AB, find a point at a given distance, d, from a given point, C.

Ex. 432. In a given line, AB, find a point at a given distance, d, from a given line, CD.

Ex. 433. In a given line, AB, find a point equidistant from two given points, P and Q.

Ex. 434. In a given circumference, find a point at a given distance, d, from a given point, C.

Ex. 435. In a given circumference, find a point equidistant from two given parallel lines, CD and EF.

Ex. 436. In a given circumference, find a point equidistant from two given intersecting lines, CD and EF.

Ex. 437. Find a point equidistant from two given intersecting lines, AB and CD, and at a given distance from a given point, E.

Ex. 438. Find a point equidistant from two given intersecting lines, AB and CD, and at a given distance from a given line, EF.

Ex. 439. Find a point equidistant from two given intersecting lines, AB and CD, and equidistant from two given points, E and F.

Ex. 440. Find a point equidistant from two given points, and having a given distance from a given point, E.

Ex. 441. Find a point equidistant from two given points and equidistant from two given parallel lines, EF and GH.

Ex. 442. Find a point equidistant from two given points and equidistant from two given intersecting lines, EF and GH.

Ex. 443. Find a point at a given distance, d, from a given line, AB, and equidistant from two given points, E and F.

Ex. 444. Find a point having a given distance, d, from a given line, AB, and equidistant from two given parallel lines, EF and GH.

Ex. 445. Find a point having a given distance, d, from a given line, AB, and equidistant from two given intersecting lines, EF and GH.

Ex. 446. Find the locus of the center of a circle that passes through two given points.

Ex. 447. Find the locus of the center of a circle that touches two given lines.

Ex. 448. Find the locus of the center of a circle which has a given radius and touches a given line.

Ex. 449. Find the locus of the center of a circle which has a given radius and touches a given circle.

Ex. 450. Find the locus of the center of a circle touching a given line at a given point.

Ex. 451. Find the locus of the center of a circle that touches a given circle in a given point.

To construct a circle having a given radius :
Ex. 452. Touching a given line and passing through a given point.
Ex. 453. Touching two given lines.

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