Given a sentence in Tagalog language, translate the sentence to English language keeping the meaning of the original sentence intact.

Q: Tagalog sentence: Tatlong taon mamaya siya ay hinirang bilang instructor sa Physics sa artilerya College, itinataguyod sa Professor of Physics doon bago siya nagretiro mula sa Army sa 1898.

A: Three years later he was appointed as Instructor in Physics at the Artillery College, being promoted to Professor of Physics there before he retired from the Army in 1898.
****
Q: Tagalog sentence: Ang point M ay ang midpoint ng Arc BC sa circle c. Kaya, ang dalawang Chords BM at CM ng c bilog ay may pantay na haba. Kaya, ang chordal angles ng mga dalawang Chords ay dapat pantay-pantay, sa ibang salita, mayroon kaming < BAM = < CAM. Kaya, ang point M kasinungalingan sa anggulo panggitnang guhit ng anggulo BAC.

A: The point M is the midpoint of the arc BC on the circle c. Thus, the two chords BM and CM of the circle c have equal length. Hence, the chordal angles of these two chords must also be equal; in other words, we have < BAM = < CAM. Hence, the point M lies on the angle bisector of the angle BAC. Similarly, the point N lies on the angle bisector of the angle BAD. But, of course, the angle bisector of the angle BAD is the exterior angle bisector of the angle BAC; hence, since the interior and the exterior angle bisector of an angle are always perpendicular to each other, the angle bisector of the angle BAC and the angle bisector of the angle BAD are perpendicular to each other. Hence, < MAN = 90.
****
Q: Tagalog sentence: Doob nakapaloob sa trabaho sa pamamagitan ng Paul magpataw ng buwis at, sa panahon ng 1940's at 1950's, siya ay binuo basic matingeil teorya at marami ng kanyang aplikasyon.

A:
Doob built on work by Paul Levy and, during the 1940's and 1950's, he developed basic martingale theory and many of its applications.
****