LOCI.

THE STRAIGHT LINE AND THE CIRCLE.

PROPOSITION I. THEOREM.

Find the locus of the middle points of all straight lines which have one extremity in a given point and the other in a given straight line.

Let A be the given point, and BC the given line, and let any line AD be drawn from A to meet the line BC in any point D, and be bisected in P.

It is required to determine the locus of P for all positions of the line AD, while the point A and the line BC remain in the same fixed position.

B

A

D

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E

From A draw AE perpendicular to BC, bisect AE in the point Q, and join PE, PQ. Then because AED is a right-angled triangle,

and EP is drawn from the right angle to the bisection of the hypotenuse,
the line EP is equal to AP or PD.
And in the triangles APQ, EPQ,

the sides AQ, QP are respectively equal to EQ, QP,
and the base AP is equal to the base EP,
therefore the angle AQP is equal to the angle EQP,
and these are adjacent angles,

therefore each of them is a right angle ;

wherefore the line PQ is at right angles to AE at Q, the middle point of the distance of A from BC.

Hence the point P for all positions of the line AD is situated in PQ or PQ produced.

Wherefore the locus of P, the middle point of AD is a straight line parallel to BC, drawn through Q the middle point of the distance of the given point A from the given line BC.

PROPOSITION II. PROBLEM.

Having given the base and vertical angle of a triangle, to find the locus of the intersection of the perpendiculars from the three angles drawn to the opposite sides.

On the given base AB describe a segment ACB of a circle containing an angle equal to the given vertical angle of the triangle, and complete the circle.

Draw AC, BC to any point Cin the circumference;
then ABC is one of the triangles.

Let the perpendiculars Aa, Bb, Ce be drawn to the sides BC, AC, AB, and intersecting each other in the point 0.

Then Cb Oa is a quadrilateral figure having the two opposite angles CbO, CaO, right angles,

therefore also the other opposite angles b Ca, bОa are equal to two right angles, a constant magnitude, (Euc. III. 22.)

but the angle b Ca or ACB is constant,

therefore also the angle boa is constant for all positions of the point 0: and the base AB is given,

wherefore the point O is in the arc of a circle described upon AB as a chord.

PROPOSITION III. THEOREM.

The locus of the centers of the circles, which are inscribed in all rightangled triangles on the same hypotenuse, is the quadrant described on the hypotenuse.

Let ABC be one of the right-angled triangles, C being the right angle and AB the hypotenuse.

Let O be the center of the circle inscribed in the triangle ABC, and let the triangle ABC be circumscribed by the circle ACBD.

Join CO and produce it to meet the semicircle ADB in D.

Then since O is the center of the circle inscribed in the triangle ABC, COD bisects the right angle ACB, (Euc. IV. 4.)

and hence also the semicircle ADB which subtends that angle.

Join AO, BO, AD, DB.

Then the angles DAB, DCB are equal, (Euc. III. 21.)
and the angles ACD, BCD are also equal; (Euc. Iv. 4.)
therefore the angle DAB is equal to the angle DCA,

and likewise the angle CAO to OAB, (Euc. IV. 4.)
whence DAO is equal to the angles CAO, ACO;
but the angles CAO, ACO are equal to AOD, (Euc. 1. 32.)
therefore the angle DAO is equal to AOD,

and the straight line DỠ is equal to DA; and also to DB. The point O is therefore in the arc of a circle whose center is D and radius DB or DA.

And since DB is equal to DA and at right angles to it, the locus of O is a quadrantal arc of the circle whose radius is DA or DB upon the given base AB of the right-angled triangle.

COR. If the triangle be an acute angled-triangle, the construction holds good, but the locus of the centers of the inscribed circle is an arc of a circle upon the base, but greater than a quadrant; and if the triangle be an obtuse angled-triangle, the locus of the centers of the inscribed circle is an arc of a circle upon the base, but less than a quadrant.

PROPOSITION IV. PROBLEM.

Given a circle and a point within it: if a straight line be drawn from the given point to the circumference and divided in a given ratio, to determine the locus of the point of section.

Let A be the given point within the given circle whose center is C, and let the straight line AB be drawn to meet the circumference in B; and let AB be divided in D so that the ratio of AD to DB is a given ratio; it is required to determine the locus of the point D.

Draw AC and produce it both ways to meet the circumference in E and F.

Join BC and draw DG parallel to BC meeting the diameter EF in G.

Then since the ratio of AD to DB is given,
the ratio of AB to AD is also given.
And AB is to AD, as BC to DG. (Euc. vi. 4.)
Hence the ratio of BC to DG is given;
and because BC the radius is given in magnitude,
therefore DG is also given in magnitude.

But AD is to AB, as AG is to AC, (Euc. VI. 4.)
Hence AC is divided in the given ratio in G,

and therefore G is a fixed point for any position of the point B. Wherefore, since the point G is fixed, and the line GD is given in magnitude, the locus of the point D is the circumference of a circle whose radius is DG and whose center is G.

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I.

5. Determine the locus of the vertices of all the equal triangles, which can be described on the same base, and upon the same, side of it.

6. Straight lines are drawn from a fixed point to the several points of a straight line given in position, and on each base is described an equilateral triangle. Determine the locus of the vertices.

7. The base of an isosceles triangle lies in a given infinite straight If line, and has one of its extremities at a given point of that line. the sum of the base of the triangle and its altitude are together equal to a given straight line, find the locus of its vertex.

8. D, E, F, G are the middle points of the sides of a rectangle ABCD. Join DF, EG; then if P be a point such that

AP + PC = BP + PD,

shew that P can only lie in DF, EG or these lines produced.

9. A square is moved so as always to have the two extremities of one of its diagonals upon two fixed lines at right angles to each other in the plane of the square: shew that the extremities of the other diagonal will at the same time move upon two other fixed straight lines at right angles to each other.

10. ÅB, BČ, DE, EF are rods joined at B, F, E, and D, capable of angular motion in the same plane, and so placed that FBDE is a parallelogram. If, when the rods are in any given position, points A, E and C be taken in the same line, shew that these points will always be in the same line, whatever be the angle the rods make with each other.

11. Two lines of given length slide upon two given lines; shew that the locus of a point, such that the sum of the areas made by joining with it the ends of the given lines is constant, is a straight line, and determine its position. Is the property true for all points of this line?

II.

12. Find the locus of the middle points of any system of parallel chords in a circle.

13. Shew that all equal straight lines in a circle will be touched by another circle.

14. If equal straight lines be placed similarly round a circle just without it, the loci of their extremities will be concentric circles.

15. If from a point in the circumference of a circle any number of chords be drawn, the locus of their points of bisection will be a circle.

16. A is a fixed point in the circumference of a circle, BC any chord, BP and CP are drawn making angles with AB, AC equal to those which BC makes with these lines. The locus of P is the diameter through A.

17. If in a circle two chords of given length be drawn so as not to intersect, and one of them be fixed in position, and if the opposite extremities of the chords be joined by two lines intersecting within the circle; then the locus of the point of intersection will be a portion of a circle passing through the extremities of the fixed cord.

18. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersections is a circle.

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19. Pis any point in a semicircle whose diameter is AB, AP is produced to Q so that PQ is equal to PB; find the locus of Q.

20. ACB is the diameter of a circle, CP, CQ are perpendicular radii, shew that the locus of the intersection of AP, and BQ is a circle whose center is in the given circle, and radius the diagonal of the square on the radius.

21. If through two given points in the circumference of a circle pairs of equal chords be drawn, one set of their intersections will lie in a diameter of a circle and the other in the circumference of a second circle, passing through the given points.

22. If from any external point any number of straight lines be drawn cutting a circle, find the locus of the middle points of the chords thus formed.

23. Find a point without a given circle from which if two tangents be drawn to it, they shall contain an angle equal to a given angle, and shew that the locus of this point is a circle concentric with the given circle.

24. Determine the locus of the extremities of any number of straight lines drawn from a given point, so that the rectangle contained by each, and a segment cut off from each by a line given in position, may be equal to a given rectangle.

25. If from a given point S, a perpendicular Sy be drawn to the tangent Py at any point P of a circle whose center is C, and in the line MP drawn perpendicular to CS, or in MP produced, a point be always taken such that MQ = Sy, then the locus of Q is a straight line.

III.

26. If one side of a triangle be constant and the difference of the squares described on the other two also constant, the locus of the vertex is a straight line.

27. Given the base and sum of the squares of the sides of any triangle, find the locus of its vertex.

28.. When the vertical angle and the sum of the sides of a plane triangle are given, prove that the locus of the middle of the base is a line given in position.

29. Find the locus of the vertex of a triangle, having given the radius of the inscribed circle and the difference of the sides of the triangle.

30. The base of a triangle and the radius of its circumscribing circle being given, find the locus of its vertex.

31. If on a given base a triangle be described such that a straight line drawn from the vertex to a given point in the base bisects the vertical angle, shew that the vertex will generally lie in a certain circle. What will be the locus, when the given point is the center of the base?

32. If a series of triangles be described upon the same base, the perpendiculars drawn to the sides of each triangle from their middle points will intersect in a certain straight line.

33. Given of any triangle the base, and the point where the line bisecting the exterior vertical angle cuts the base produced, find the locus of the triangle's vertex.

34. Two fixed straight lines AB, AC are cut by any straight line